This is a great question! Other countries in Europe, England for example, are a different matter, but I can answer it with regards to the Lutheran tradition in the German lands. The education of clergy was one of the bigger concerns that Luther and his compatriots faced from the late 1520s and especially in the 1530s and beyond - from the posting of the 95 Theses in 1517 up until around 1527, there were still hopes of dialogue and reconciliation with Rome, but as the controversy dragged on year by year, they started to come to terms with the need to figure out these systemic questions of church organization and functionality.

I say 1527 because it was in that year that Luther undertook a comprehensive visitation and evaluation of the parishes in Saxony, and he was appalled at what he found. In his preface to the Small Catechism, he wrote:

The deplorable, miserable conditions which I recently observed when visiting the parishes have constrained and pressed me to put this catechism of Christian doctrine into this brief, plain, and simple form. How pitiable, so help me God, were the things I saw: the common man, especially in the villages, knows practically nothing of Christian doctrine, and many of the pastors are almost entirely incompetent and unable to teach... Oh, you bishops! How will you ever answer to Christ for letting the people carry on so disgracefully and not attending to the duties of your office even for a moment? One can only hope judgment does not strike you! You command the Sacrament in one kind only, insist on the observance of your human ways, and yet are unconcerned whether the people know the Lord's Prayer, the Creed, the Ten Commandments, or indeed any of God's Word. Woe, woe to you forever!

Behind Luther's rhetoric lies an important point: a significant change in the way the office of the clergy was viewed before and after the Reformation. Before, the role of a priest was seen as primarily sacramental, providing the seven sacraments to the people - and most of all, the Eucharist and saying Mass. For the Reformation, the role was itself much more educational, with Luther's focus on teaching Christian doctrine and Biblical content to the laity. Luther concludes:

So look to it, you pastors and preachers. Our ministry today is something else than it was under the pope.

In the years immediately before the Reformation, the clergy in prominent positions and prestigious parishes would often have a university degree. Luther himself went through a Latin grammar school as a child, entered the University of Erfurt at age 17, and received bachelor and master of arts degrees there. He entered the monastery in 1505 before being ordained in 1507 - but notably, it was after his ordination that he was sent back to the university to acquire a specifically theological education, bachelor's degrees in Biblical studies and theology in 1508 and 1509 before finally receiving his Doctor of Theology in 1512. While he was already educated and literate before his ordination, his theological education largely came afterwards.

A contemporary Reformer, Ulrich Zwingli in Switzerland, had a somewhat similar experience: he received a Master of Arts from the University of Basel before being ordained, yet had very little theological knowledge. During the ten years at his first parish at Glarus, he continued his self-study in Greek and Hebrew, theology, and humanist learning. This was not particularly noteworthy for the time, for a man to be ordained with little theological expertise; the focus of priesthood was first and foremost on saying Mass, and many rural priests were more or less apprenticed to an older priest and taught that way, with no real formal education involved. Even these educated and prominent priests like Luther and Zwingli were ordained before their theological education really began.

So all this is to show that before the Reformation, the training of priests was not always a rigorous or standardized affair, and Luther found many rural village priests to be essentially lacking in what he considered rudimentary Biblical and theological knowledge. In the late 1520s, Luther wrote two catechisms, the Small and the Large. The Small Catechism was primarily for laity, and according to his preface, especially for a father or head of the household to teach the rest of his family about basic Christian doctrine. The Large Catechism covered the same general topics but in somewhat more depth, and it was a crash-course textbook for parish pastors to learn the basics of the faith in accordance with Reformation theology.

Now, given Luther's emphasis on education for all, it was suddenly a much higher priority for clergy to be sufficiently educated. One of the places this happened in the earliest days was at Luther's feet: the University of Wittenberg saw explosive growth in the number of enrollments with Luther was the superstar professor. Further Protestant universities were founded at Marburg in 1527 and Koenigsberg in 1544. Sometimes it was even literally at Luther's feet; he and his wife often hosted young students, with their house an unofficial dormitory, and his friends and students would father after dinner over a beer (or few) to discuss theology, the Bible, current events, and so forth. Many of these conversations were jotted down by his students and have been published as Luther's Tabletalks.

However, it was Luther's right-hand man Philip Melanchthon who did one of the single most important things for theological education: he wrote the first Reformation work of systematic theology, the Loci Communes. One of the reasons Luther never wrote a systematic theology himself was that he praised Melanchthon's Loci highly and saw no need to reinvent the wheel. This largely took the place in Lutheran education of Peter Lombard's Sentences, the standard work of theological education for several centuries before. The Loci was originally published in 1521, with major updates, expansions, and revisions in 1535, 1543, and 1559 right before Melanchthon's death.

Melanchthon also is known for his legacy as the "Preceptor (teacher) of Germany" for his work in creating new educational materials and institutions for Saxony in particular and Germany in general. The Lutheran tradition was from the beginning a strong proponent of universal elementary education (most forward-thinking, for girls as well as boys) but one motive for this was certainly to train new leaders for the church. Luther wrote that one of the main purposes of these schools was "for raising up people who are skilled to teach in the church and govern in the world," and the Lutheran teaching of vocation regarded the pastoral office as an honorable path for children to consider and pursue.

So to summarize/recap, before the Reformation the standards and methods of training for priesthood was somewhat haphazard; educational reform was a major priority for the early Reformers. With their new catechisms and other materials they sought to educate pre-existing clergy; with their new universities and theological texts, they sought to raise up new, well-educated and competent pastors.

Sources:

• The Small and Large Catechisms, Martin Luther

• The Book of Concord, Kolb-Wengert edition

• Brand Luther, Andrew Pettegree

• Commonplaces (Loci Communes), Philip Melanchthon

• Historical Introductions to the Book of Concord, Friedrich Bente

Okay since we were early reformers here in Sweden, let's go..

Background

So it's 1515 or so. Sweden's nominally in a personal union with Denmark under the rather autocratic Christian II of Denmark. De facto the country is ruled by the elected Regent Sten Sture the younger. He and his followers are in open armed rebellion against the union, which means he's demanding allegiance and at odds with pro-union nobility and not least the clergy, which has a lot of pro-union sympathies. The church at this point in time is wealthy and powerful after reciving tithes for a few centuries; together the bishops have more wealth and more castles than the crown does, and also control entire counties (län). So archbishop Trolle, a pro-unionist gets back from Rome (having stopped and met with Christian II of Denmark on his way) and finds Sture's confiscated his county. The situation escalates from there; Sture demands an oath of loyalty from Trolle, and lays seige to his castle at Stäket (one of four the bihop had, together with Arnö, Tuna and Uppsala) in the fall of 1516. Mediation was attempted to no avail. Christian invades to try to liberate Trolle twice, but is repelled by Sture's forces. By late 1517 Trolle surrenders and quits. The Riksdag (parliament) passes an act authorizing the destruction of the castle, and it is. The clergy were not unified in supporting Denmark though; The bishop of Strängnäs Mattias Gregersson is one of the signees of the act.

Meanwhile in 1518 a young renaissance humanist named Olaus Petri returns from his studies at the university of Wittenberg and starts working for the the aforementioned bishop Gregersson. In Wittenberg Petri had been there to witness Martin Luther's nailing of his theses to the church door and had become an early and strong suspporter of Luther's ideas, which he promotes in Sweden.

Trolle goes down to Denmark and Christian II invades again in january 1520; and meets Sture's forces in battle on the ice of a lake. The third time apparently being a charm for Christian, a cannonball ricochets off the ice, killing Sture. His troops advance across the country and in March the Swedish national council declares Christian king of Sweden. Sture's wife Kristina Nilsdotter and Sture loyalists remain in power in Stockholm though, and being a proper medieval city with walls, the siege continues until the fall when the city falls.

Christian crowns himself king, reinstates Trolle as archbishop. Immediately after the coronation party Christian starts an 'inquisition', using the act of parliament as evidence he accuses the Sture supporters of heresy and has upwards of 100 of them executed. Bishop Gregersson's head is the first on the block. This is the 'Stockholm Bloodbath'. (also called that in Denmark) Olaus Petri likely witnessed this too.

It turns out executing a bunch of VIPs for no good reason, when they have large popular support isn't a good move. This really turns the fight into a more purely nationalist struggle of Sweden vs Denmark. Even much of the Danish population consider this crossing the line, not least the Danish nobles who don't want a precedent set. When Rome starts poking around asking what that 'inquisition' they'd never sanctioned was all about, Christian II scapegoats his archbishop Slagheck and has him burned at the stake as a heretic two years later.

Anyway a young nobleman named Gustav Vasa, who'd had a bunch of his relatives killed in the bloodbath, immediately starts his own uprising. Within a year he's been declared Regent and - yadda, yadda yadda - in 1523 he's king of Sweden. With some help from Lübeck, who Christian II was also picking a fight with. Another archbishop, Johannes Magnus, is elected - still Catholic but one with more nationalist sympathies and loyalty to the king.

The reason for all this background, is to show the very difficult situation the Catholic Church had in Sweden at that point in time.

Vasa gets to work

We have a new king who wants to expand his power, who's rode into power on an uprising with both strong nationalist and anti-clerical sentiment against the archbishop and the wealth of the church. This at a time when Luthers ideas have started to spread in the country, and stoking attitudes against the wealth of the church. Vasa gets to know Laurentius Andreae from Strängnäs and is taking an interest in his Lutheran ideas. Through him he also meets Petri, who by that point has become a deacon and is busy preaching Luther's Christianity in Stockholm. Other clergy are starting to openly preach in Luther-inspired ways.

In a move that can't be described as anything but politically dumb, the Vatican stands by their man, insisting that Trolle, exiled to Denmark again, is still the rightful archbishop of Sweden. Needless to say, Vasa doesn't want him as archbishop. He wants him dead, executed for treason. Rome finally confirms the new archbishop in 1524 but the damage may have been done. If Vasa hadn't decided to embrace the reformation before he was king, it wasn't long after. But he was careful. Rather than choose one of the reformers, he tasks his Catholic archbishop with translating the Bible. However, Petri is the guy who ends up completing the job and the New Testament is published in Swedish in 1526. He's also publishing other reformation literature. For Christmas that year, Vasa suggests a public debate between Olaus Petri and the catholic professor of theology at Uppsala University, Peder Galle. Galle declines on the grounds that he's not authorized to speak for the church, and that the doctrine of the church is not a matter of debate anyway. Thus, Vasa scores a PR victory for Lutheranism without having to look like he's advocating it.

In 1527 Vasa gets the Riksdag to confiscate a massive amount of church property, including all bishop castles, as well as banning bishops from having either castles or retainers and declaring clergy subject to secular law. At the same time the 'the pure word of God' is to be taught and preached. In practice this amounts to an official go-ahead to preach Lutheran ideas, but still has plausible deniability. This leads to some defections among the more ardent Catholic clergy; including the Bishop of Linköping who goes into exile. This only has the effect of strengthening the reformation though, there was no shortage of people who'd like to move up to bishop. (or is it a move diagonally?)

Ideas spread and the big break comes in 1531. Vasa appoints Laurentius Petri - Olaus' brother and an equally convinced Lutheran - as new archbishop, and Laurentius is anointed as such by the (nevertheless Catholic) bishop of Västerås. Hence the Church of Sweden claims to maintain an unbroken line of apostolic succession, as the Anglicans do. Not that the Vatican would agree. An increasing amount of church literature is being written in the vernacular, such as manuals. In a rather Stalinesque series of events, Olaus Petri would gradually fall out of favor in the '30s and was ultimately sentenced to death for high treason in 1540, the sentence commuted and then he was brought back in favor. Vasa being skilled at politics doesn't mean he wasn't a despot.

At this point in time the church would have been more accurately described as reformed-catholic than Lutheran. The final break with Rome was in 1536 when the church council at Uppsala rejects Canon Law. The aforementioned Peder Galle was likely there; in other words, there remained a significant breadth of opinion in the church, from Catholics to reform-minded-Catholics to Lutherans. It wasn't until 1593 that the Augsburg Confession was officially made church doctrine. It was a bit of a 'big tent' for a while, even if some historians (Lindqvist) describe it as becoming 'fundamentalist Lutheran' by the next century.

Some theological details had to be sorted out, too. A medieval debate on whether expensive imported wine was required for the Lord's Supper was rekindled - because if transubstantiation wasn't physically real, did the wine have to be? (they came down on the side of 'yes' in the end, but not before some frugal priests had been experimenting with lingonberry juice,etc!)

So, the answer is:

By and large the clergy was neither replaced nor re-educated. However the Church of Sweden is a 'high church' denomination, in that it's as close to Catholicism as any protestants other than the Anglicans perhaps. There's apostolic succession, no lay priests; they studied and continue to study theology at the same institutions as before, but with the Lutheran doctrines. Some Catholic sacraments are maintained (e.g. Confirmation and the Lord's Supper are part of it while Confession and Last Rites are not), and there's still a church hierarchy.

Sources

Err, well this is such a pivotal event in the country's history that no Swedish history book covering the period doesn't go through most of the above. I can recommend some on request.

It's old (and therefore digitized online) but decent history of the church in greater detail is

Carl Cornelius, Svenska kyrkans historia efter reformationen, Förra delen (1520-1693), as a starting point anyway. If you want more depth and angles, there's like a hundred years of Kyrkohistorisk tidsskrift to plow through.

N.B., by the way: I'm glossing over here the popular resistance to the reformation, which absolutely did exist, and just as the reformation itself was fueled by complicated political factors, so too was the resistance complicated by general dissatisfaction with Vasa policies. These were strong enough in some parts of the country (Dalarna, Småland) to lead to outright uprisings. But for the most part people were eased into it. True to form, Vasa would write a letter to Dalarna to the effect that he no plans of any sort to change religion in the country, "but now that you mention it, surely one can praise the Lord as well in Swedish as in Latin?"