I'll mostly address the theological aspect, since that's my concentration. But yes, it did happen pretty much right away: if we date the beginning of the Reformation to Luther's posting of the 95 Theses in October 1517, it was within the first few years that Luther began raising doubts about all of it.

I would say that on the part of Martin Luther and his fellow Reformers, as the theological leaders of the Reformation, the question of venerating saints was a theological concern, not a social or practical one. Luther's big problem with it was that the cult of the saints, in his view, put up unnecessary barriers between the person and God and, too often, degenerated into idolatry in which another human being is loved and trusted more than Christ.

Luther began reacting against this and rejecting it as early as 1521-1523, first stating that he found no Biblical basis for the practice and therefore he could not condemn anybody who declined to venerate the saints, then later actively rejecting such veneration as idolatry:

For about the intercession, adoration, and prayer to departed saints there is nothing in Scripture. No one can deny that by such saint worship we have now come to the point where we have actually made utter idols of the Mother of God and the saints, and that because of the service we have rendered and the works we have performed in their honor we have sought comfort more with them than with Christ Himself. Thereby faith in Christ has been destroyed.

That's not to say that it was his intention to totally banish all mention of the saints. There are differences with other Protestants, but the earliest Protestant view and still the current Lutheran is that,

the saints are to be remembered so that we may strengthen our faith when we see how they experienced grace and how they were helped by faith. Moreover, it is taught that each person, according to his or her calling, should take the saints' good works as an example... However, it cannot be demonstrated from Scripture that a person should call upon the saints or seek help from them. "For there is only one single reconciler and mediator set up between God and humanity, Jesus Christ" (1 Tim. 2:5)... He alone has promised to hear our prayers.

From the Augsburg Confession, XXI. This dates from 1530, the first systematically organized Protestant/Lutheran confession of faith. This is a fairly consistent thread in Lutheran thought: it's fine (even encouraged) to remember the saints, to honor them, to learn from their example. What crosses the line is actively praying to them, to look to their merits (see the "treasury of merits" connected to indulgences) instead of to Christ, etc. It's the same in much of the theology connected to the Virgin Mary: it seems that throughout his life, Martin Luther continued to believe some very Roman Catholic things about Mary, including her perpetual virginity and even her immaculate conception. These doctrines weren't the real problem for him; the issue was the active devotion, intercession, and mediation.

Of course, even if the Reformation leaders were primarily motivated by theological concerns, there were certainly other non-theological factors involved with their followers. The cult of a popular saint who drew a lot of pilgrims was a huge economic boon to a city or region that possessed his or her relics; the thriving Medieval trade in relics testifies to this, as different places competed and sought to one-up each other in the relics they possessed. So the theological opportunity in the Reformation to dismantle this would be appealing to places that suffered as others gained - a way to level the playing field, so to speak. One particularly popular destination was Santiago de Compostela in Spain, in addition of course to Rome itself, and the Germans were all too conscious of all the gold leaving their own land, going into the pockets of the Spanish and Italians. Luther mentioned this too, as early as 1522:

We read no more about St. James in Acts than that Herod killed him with the sword; that is all. How he ever came to Santiago de Compostela, whither people undertake the great pilgrimage, we are not reliably informed. Some say that he lies in France at Toulouse, but they too are not sure of the matter. Therefore let the saints lie, and do not run to their shrines; for you do not know whether St. James or a dead dog or a dead horse lies there. Therefore those who run there are properly served; for since they neglect to do the good, right works, which God commands, they fall into running to St. James and, before giving a poor man ten gulden, they run to the shrine and use up forty or a hundred.

One could speculate on this shift in the 16th century being related to the changing shape of society as a whole. In his book The Cult of the Saints, Peter Brown makes the case that these devotions emerged at least in part due to the social importance of a patron-client relationship; the saints were seen as "friends of God," so friendship with the friends of God was a valuable patronage to gain, just as friendship with the friends of a king or emperor would be. This plays in well with the Medieval feudal system as well, but as that faded into the early Modern era and structures change, the perceived need in that regard would be lessened. That's not my wheelhouse, though; I'd love to hear a social historian comment on that.

Dismantling the cult of the saints is, I think, one of the most fascinating developments in Reformation piety. With most things--right down to something as central to Christianity as the Eucharist--we see adaptation and reinterpretation, not outright destruction.

And it was a more radical move than Luther tended to favor on his own. /u/Philip_Schwartzerdt places the emphasis differently than most religious historians would today. In the early 1520s, indeed, Luther was waffling on the cult of the saints. (He himself had a particularly strong devotion to St. Anna, Mary's mother). This is a man who made NO secret of his iconoclastic (literally and metaphorically) beliefs in writing and NO attempts in the treatises in question to be accommodating or light-handed. So when Luther suggests that some people might not venerate the saints and it's okay, he's still accepting the cult (that is, organized devotion, Latin cultus) as valid doctrine. In fact, the pressure to reject it altogether came from more radical elements among the Reformers, and its adoption by Luther, like his reluctant decision to give up his personal celibacy, seems to have been largely pragmatic.

So why did people want to get rid of the saints, and why were they religiously able to dump such a MASSIVE component of late medieval devotion?

I want to point to three key functions of saints in medieval piety. First, they were intercessors with Christ for mercy and salvation after individual Christians' death. Second, their pious lives were exemplary--they were role models. Third, they were "holy helpers"--as patron saints of particular causes and cities, they intervened in daily life to help people in this world, not just the next.

Melanchthon has properly covered the first aspect: Protestant soteriology (theology of salvation) has no room for saints. Salvation depends solely on Christ, and is predestined for each individual. Prayers from a saint have no practical impact. And the idea of people "more graced" than others flies in the face of the Protestant "priesthood of all believers" ideal.

But what of the other two? Those don't really counteract any Protestant dogma--in fact, they're pretty important to it. Christians today still seek role models of morality and piety; they still pray for help in this life.

The thing is, people had spent the last century and a half of the Middle Ages entrenching alternate or competing methods of filling those two purposes.

First, while biblical exemplars had always been hanging around Christian teaching, references to Bible heroes and heroines for role models became much more prominent and frequent in the 14th-early 16th centuries. Judith is everyone's favorite example (even though the book is apocryphal). In the hands of male authors, she models chastity, purity, good widowhood. In the hands of women authors, she embodies brave heroism in the face of trials. (For reference: Judith in the Hebrew Bible deuterocanon is an Israelite leader who cuts off a dude's head with a sword.) During the 15th century, more and more people had access to the Bible, and of course Reformation preaching centered on it. People still had their role models, even without the saints.

Second, and way cool IMHO, is the matter of intercession in this life. Keith Thomas can talk about the "decline of magic," but we're not quite there yet in 1520. Women are still writing scraps of St. Margaret of Antioch's hagiography onto amulets for protection during childbirth. But from the early 15th century, propogated by popular preachers/didactic writers like Jean Gerson, there was a new cult arising: guardian angels. The idea of an angel, not spectral human, helper for individual Christians already had solid traction in popular piety by the time the Reformers rolled onto the scene.

So we've accounted for the main purposes of the saints in European Christianity circa 1520. One prong was theologically unnecessary, outright contradictory. The other two had ready substitutes already well established in people's mind. That's the setup. For the virulent opposition that either made Luther change his mind or pushed him in a direction he was tentatively leaning, we'll turn to the nature of lay anticlericalism in the early Reformation. Well, one tiny sliver of the nature of lay anticlericalism, heh. And that is: the cult of the saints was a central component of what was perceived as the financial abuses of the Roman Church. "Donations" to see saints' relics; pilgrimages to saints' shrines to obtain desirable indulgences--popular saints were a real moneymaker for the Church. And again, lay nobles had spent the 15th century trying to reform the fiscal abuses of their local parishes in their territories; the entrance of evangelical (Lutheran) preaching and its anti-Rome anticlerical message unleashed massive floodgates. The cult of the saints, with no particular reason to exist, could not as an institution hold up under the anger reflected onto it.

Scholars have shown, of course, that individual and local pockets of veneration continued into the Protestant era. Changes aren't immediate. But the improvement and spread of Protestant education, and the substitutionary roles of guardian angels and biblical exemplars, ultimately filled in the gaps in the wall.