They did end up being quite visible in Swedish history to this day even, since a lot of them were ennobled and started dynasties (e.g. Douglas, Hamilton, von Wright, Spencer, Craaford)

Scottish mercenaries had already been showing up in the otherwise German-dominated market for mercenaries in the Baltic in the preceeding century. An early note is found in the Tankebok of Stockholm for 1520. Scottish mercenaries left there by Christian II of Denmark after the Stockholm Bloodbath had apparently been causing trouble, as their leader 'provost Jacob' was called up to the city council to swear to harshly punish any of his men that molested any burgher, rich or poor. Recruitment spiked in the Livonian War, and at the end of the 17th century there were about a half-dozen Scotsmen commanding Swedish castles or garrisons.

By the 17th century, the way the recruitment of mercenaries worked, was through an agent with a 'recruitment patent' (värvningspatent). This would typically be someone appointed to command a regiment - provided they could recruit it. One of the most significant of those was James Spens, a Scottish adventurer who'd entered Swedish service already under Charles IX, in 1606, another was Patrick Ruthven, who came a few years later around 1609. While Spens was an adventurer who'd previously failed as a planter on the Isle of Lewis, Ruthven belonged to the family of the earls of Gowrie, who'd just been disgraced after the Gowrie Conspiracy; so going into exile was a gambit to preserve his family name and status. Yet another fellow is Alexander Leslie, who'd been already serving for the Dutch when he was recruited there, and there was a lot of recruitment among Scotsmen in the Netherlands. A similar story was Donald Mackey, recruited from Denmark (foreign recruitment focused on these nearby parts such as Stralsund, NW Germany, Denmark etc)

All these men would be tasked with recruiting one or more regiments of Scotsmen. Some recruited included James Seaton ('Cornelius James' in a Swedish source, possibly a corruption of his rank in English - Colonel) and James Ramsay ("the Black", to distinguish him from another James Ramsey "the fair" leading the Marquis Hamilton's regiment, which was raised by Hamilton himself and not the Swedes). These men in turn raised further regiments in the Thirty Years War. to some extent it could be viewed as a case of chain migration, or a 'business cluster'.

British participation was not limited to Scots though; Spens went to England in 1629 to raise two regiments. It's not certain who led one, the other was lead by George Crafoord and at least that one was considered mismanaged by Lord High Councillor Axel Oxenstierna, due to friction between the English soldiers and their Scottish officers. Irishmen had also been recruited by Scots earlier, under Charles IX - some of them likely pressed into service, and for that reason had a tendency to desert at the first good opportunity. So because of that and their adherence to 'papism' made the Swedes order the Scots not to recruit Irishmen. So, Grosjean lists over 90 Scottish colonels in Swedish service 1629-1648 but only 7 English and Irish.

Germans would nevertheless remain largest recruitment pool.

Sources:

Alexia Grosjean, An unofficial alliance: Scotland and Sweden 1569-1654, 2003 (pretty much this whole book is on the topic, so..)

Jan Glete, Värvade Regementen i Svensk Tjänst 1618-1631

Just for fun; it's not really a source here but a contemporary English insight in the rules and lives of the soldiers in Swedish service:

William Watts, The Swedish discipline, religious, civile, and military, 1632