I thought it is important to clarify the question pointing out that in EU4 recruiting mercenary regiments is done one at a time per controlled province. So the short time for tens of thousands (that would suggest the minimum is at least 20.000 mercenaries) might not be so short if player is OPM (one province minor) or impossible since mercenary pool is limited and depends on amount of controlled provinces. So for small or medium sized countries getting 20.000 mercs could easily take 3- 6 months of in game time even if they have mercenaries and gold available.

First off, apologies for being late to the party, but I’ll hopefully be able to add on some useful stuff here.

Now, as much as I love EUIV, it does misrepresent armies a little bit. Throughout most of the early modern, standing armies weren’t really prevalent. I can’t speak as well for other states, but within the Habsburg Hereditary Lands (Austria, Bohemia, Royal Hungary) the only ‘standing army’--by which I mean a force on continual readiness, rather than a militia--were the borderers on the Military Frontier between the Habsburg lands and the Turks. Beyond that, defence was left to a combination of local militias controlled by the provincial estates, and mercenaries who could be hired by the Emperor.

The early modern was the era of what is often termed ‘military entrepreneurship’. The military entrepreneurs are often whom we think of as among the great names in this period, including Mannsfeld, Tilly, and Wallenstein. Theoretically, what would happen is that the monarch would commission an individual to raise a force, and continue to pay for that force and its services. However, given the relative infancy of institutions for state collection of money during the early modern, often what would happen instead is that the entrepreneurs would raise the armies on their own resources, pay the army in the spoils of war, and--at least in the case of the Imperial entrepreneurs--gain heavily in power/favour, along with acquisition of titles and lands. Paying for the war was a massive problem, and a huge burden on the finances of almost every party involved in war during the early modern.

Now, this gets to the meat of your question: How quickly could such regiments be raised, and where did the recruits come from? I’ll answer the second one first, as I can offer a more concrete answer, whereas the former will be a bit more vague.

The recruits came from everywhere. They came as deserters from the opposing army, who left seeking better pay, conditions, or looting opportunities in the service of another general. They came as volunteers from the area where the army was operating. They came as recruits mustered by local militia levies, and as foreigners come to join the opportunities for looting present in war. They joined because the army offered work where otherwise there might not be any, or to escape their present conditions, and others were effectively conscripts, either forcefully enlisted or sentenced to service.No doubt some came because they were professional soldiers, and war was their career. There was definitely a mix of volunteers who wished to join the army, and other who were--in some way--compelled to join.

These definitely weren’t ‘national’ armies as we would think of them today. A Frenchman might just as well fight under Spinola for Spain, then desert to for the Protestants under Mannsfeld. There would be regiments of Irish and Scottish extraction fighting in an army raised for the Emperor--indeed, Wallenstein was killed by Scottish and Irish colonel’s of his army. The Spaniards in particular would raise regiments from one part of their empire to fight in another.

As for how quickly they could be raised, it’s a bit harder to tell. Wallenstein was authorised to raise an army 24,000 strong in April 1625, and had a force of 16,000 by the beginning of 1626. However, over 1626, he managed to raise an impressive force of 70,000 total, with about 35,000 in the field army under Wallenstein. However, this is from the middle of the Thirty Years War, and so can’t really be taken as fully exemplary of the period. If I were to hazard a guess, I would say that EU IV runs things a bit too quickly. It definitely took time both to marshal the money necessary, recruit the men, equip them, and muster them in a state ready for action. While I can’t give a precise time, perhaps it is telling that it was two years between the beginning of the Bohemian Revolt and Battle of White Mountain.

In any case, I hope this goes a way to answering your question. Feel free to ask any follow up questions you may have.

Sources

Michael Hochedlinger, Austria’s Wars of Emergence

Geoffrey Parker, The Military Revolution

Peter H. Wilson, The Thirty Years War: Europe’s Tragedy