It is not clear just how many soldiers of the Ottoman army were captured during the Battle of Vienna, but they probably numbered in the thousands, and many more would follow in the years and decades after, while the centuries-long intermittent and war between the Habsburg realms and the Ottoman Empire raged on. The two opponents employed similar tactics when it came to POWs and other captives of war, and both captured and kidnapped countless men, women and children throughout the Early Modern Era, taking them far from their home to whatever fate would await them there. Their individual destinies are highly varied, and since they ended up all over the two empires over quite a long period of time it is hard to make any collective statements about them. There has been some (mostly localised) research on that topic and on individual lives, however, and I'll try to write a bit about that in the hopes that this will be representative for the captives of Vienna, even if I'm not aware of any scholarly work that tries to track specifically them.

Maybe a quick word on the terminology first: If you read up on that topic, you need to be aware of the changed meaning of the word "Turk", which in early modern Germany commonly was used for any Muslim subject to the Ottoman sultan and therefore also included e.g. Bosniaks and Bulgarians.

There have been "Turkish" captives in Germany for a very long time. The first one we know of was Sadok Seli Soltan, who was captured and taken as a slave by a German lord during a crusade in the late 13th century and taken to Germany, where in 1305 he converted to Christianity, took the name of "Johannes Soldan", settled down and married. He might even have been an ancestor of Goethe! From the 16th century onward, when the Habsburgs were increasingly often involved in war against the Ottomans, the number of "Turkish" slaves in Germany and all of Europe rose steadily; another source of slaves was the fight against Muslim slavers and pirates in the Mediterranean. After the 1683 Battle of Vienna, their number skyrocketed, and over the next decades thousands of Turkish POWs and other slaves were forced to go to Germany. For the most part, the captives remained either within the warzone or in the areas adjacent to it; those who were wealthy or otherwise important were ransomed and more often than not got to go home relatively soon, while those whose families couldn't pay the ransom were forced to work as slaves on farms or in a military context. Others were given to the Spanish and French navies as galley slaves. Only a relatively small percentage of the non-wealthy captives ended up being sent off to Germany and other parts of the Empire; amongst them were disproportionally many women and children.

These slaves were taken for a number of different reasons. For one, they represented a well of manpower that could easily be used for larger enterprises; we know for example, that many Turkish captives of the 1683 battle were initially forced to take part in the cleanup efforts in Vienna, freeing the streets of bodies, debris and even unexploded ordinance, while 350 slaves taken by the Duke of Bavaria in Buda were at least partially forced to work at the construction of Schleißheim palace. Others worked in agriculture, in mines or as sedan chair carriers. Another common thread was the desire of victorious soldiers and officers to bring a "trophy" back home, and one of several slaves were seen as especially prestigious in that regard. There was also a theme of Christian Europeans (especially nobles) connoting "Turks" with an exotic nature, and they were therefore often taken to Germany as a curiosity to be shown to guests. The possibility to exchange those captives for Christian captives taken by the Ottmans was very important especially to the Christian clergy of the time. And lastly there was also a simple motive of profit, since Muslim slaves could either be sold off or maybe yet sent back to their home for a monetary ransom.

Slavery in early modern Europe wasn't comparable with the systems of cattle slavery found in the American colonies. Since the Muslim slaves ended up all over the Empire and beyond and never were numerous enough to become an economic factor of their own, they often were employed as servants by the rich and powerful, be it as "decorative slaves" to be shown off to guests, as domestic servants, in the stables or elsewhere. Another consequence of the slaves being spread out all over the Empire was that there never developed a common identity of "Turkish slaves"; and lastly, there seldom was any incentive by their masters to keep up their status as slaves indefinitely. Instead, many Muslim slaves eventually ended up as free men and women. Some of them wound up going back home, many other chose to stay, becoming an integral part of their new home.

The individual stories could be quite different. While many of the (former) Muslim slaves ended up converting to Christianity, many other never did, living their life as isolated Muslims in the European diaspora. "Turks" were found not only all over the Empire in large cities, royal residences and small towns, but they were also found in all walks of life from servants at the royal table to innkeepers, confectioners, farmers, soldiers, musicians and even monks and priests. Most of those who stayed eventually married a Christian man or woman; Muslim-Muslim marriages were exceedingly rare. Some of the Muslim captives even were raised to nobility! The extant sources seem to show that, once freed, they were on the same legal footing as everybody else, no matter whether they did convert or not. Muslims or former Muslims can be found signing court documents, petitioning the court and, in one case, even suing the King of England. That doesn't mean that it was all great, however; sources indicate that trauma from the experiences of war and captivity was widespread (a common occurence was the tearing apart of families, e.g. when shortly after the Battle of Vienna a pregnant Muslim woman was sold in Leipzig to a merchant for 50 kilograms of sugar, while her 6yo son was bought by two other merchants for ten thalers), and in some cases there might also have been active religious discrimination against them both before and after their being freed, e.g. when Muslim women slaves were forced to work with pigs and pork. On the other hand, those same women were allowed to write up and send a petition to the sultan in which they asked him to buy them back, which again shows that the enslavement of Muslims in Europe wasn't necessarily seen as a permanent thing. In Bavaria, several dozen "Turks" managed to escape and flee their captors; some of them later came back, however, and asked to be put to work again. Many slaves had to put in a lot of effort to learn German, while at the same time they couldn't expect anyone in their new home to understand whatever language they spoke natively. One Turkish woman who ended up in Silesia got lucky: She spoke an unnamed "Slavic language" (maybe Serbo-Croatian or Bulgarian) and therefore was able to (poorly) communicate with those in her new home who spoke Polish. There is also at least one incident of a Turkish slave ultimately committing suicide.

Still, the common thread for many or even most of those who ended up in the Empire (which, again, was only relatively small minority amongst the total number of captives) was eventually integration and a new normalcy. In some cities where a larger Muslim community came to be after 1683 (although even the largest communities never numbered more than a couple hundred), Muslim cemeteries and even small prayer houses were constructed for those, who had not (yet) converted to Christianity. Those could even be found in ecclesiastical territories like the prince-bishopric of Würzburg. On the other hand, the baptism of "Turks" was used as a great spectacle, through which the superiority of the Christian faith was to be ceremonially showcased. It is naturally hard to say how many of those baptims were motivated by true belief, and how many were used as a tool to speed up the integration process, but seeing as there was no obligation to convert, and that there were indeed former Muslims who entered Christian clergy after their own baptism we might assume that many converts really did take their new religion seriously. It's also interesting to see that women seem to have been much more likely to convert than men: After three years in Vienna, 80% of female captives and about half of the male ones either had already converted or were in the process to convert.

To sum it up: The individual trajectories of men, women and childred taken captive by Christian troops during the Early Modern Era were highly varied and different. Those who could afford it were soon sold back to their families, whereas those who couldn't ended up as galley slaves, work slaves in the frontier areas or work slaves and domestic servants in the Empire. Many of those who got sent to the Empire would eventually receive their freedom and either integrate into Christian society (often also by changing their religion, though not necessarily) or travelling back home. It is almost impossible to come up with any sort of reliable numbers, but after three centuries or more it is probably safe to say that a significant percentage of Germans and Austrians today has a former Muslim slave somewhere in their family tree.