This is my first answer on this subreddit after quite a long hiatus and I think i may have lost what little flair for answering questions I once had.

The Thirty Years War is a historic mess filled with shifting alliances and semi declared belligerents, it is important for a modern viewer to remember that the concept of 'war' as we understand it is not quite applicable here, it was possible for a country to provide financial aid and mercenaries without actually declaring war on one of the sides. It is interesting to note that during the Thirty Years War an equivalent number of British troops went to fight on the continent as were constricted by Sweden yet with almost no diplomatic impact or strategic gain.

English (and from this point I'm going to include Scottish under the title British) involvement in the internal politics of the Holy Roman Empire began in 1613 with the marriage pf Frederick V of the Palatinate and the Princess Elizabeth Stuart. By this point Frederick and other radical Protestants was convinced that there was going to be a war and that James I was forming an alliance that would stop the Emperor and his plans to crush Protestantism, James had no such plans and was instead hoping to avoid war by balancing the power of the Hapsburg's and that of the Protestant Union.

When Bohemia revolted it was expected that there would be widespread international support for the new regime (If you don't already know Fredrick was crowned King of Bohemia) but they were sorely disappointed. The Dutch permitted two regiments of British and German troops serving within their army to join the Bohemians but no more.

James still hoped to maintain peace in Europe and sent ambassadors to both sides to try to mediate. The Bohemians felt terribly let down and the Imperialists were only interested if he could restrain Frederick,it was obvious that he could not. A loan was refused but the Palatinate was able to raise enough money to hire 2500 English mercenaries, these were joined in October 1620 by an additional 2000 volunteers under Horace Vere.

After the destruction of the Bohemians at White Mountain James (and later Charles) continued to vacillate, both knew that Britain had to act to satisfy domestic pressure and maintain their prestige but were not sure how to do this, there was no way to pressure Frederick into accepting negotiation with the Emperor even though the court spent £1.44 million recruiting troops for various 'paladins' who took up the cause of the Palatinate on the continent.

If you're interested in a few figures a total of 127,950 of Britons were recruited to fight for the Protestants and 42,587 for the Habsburgs, the largest single benefactor was the Spanish who received some 32,000 troops followed by the Swedish with 30,000 and then the French with 25,000. It is worth noting the vast majority of those who fought for the Emperor or Spanish were Irish and so jointly Catholics and not English.

Between the collapse of the Bohemians and the intervention of the Danes the English were seen by the various generals who continued the Palatinates cause as the best hope for troops and financial support, Ernst Mansfeld was the primary beneficiary receiving some 13,000 troops over the course of his four-ish year struggle but this was far less than what was supposed to have arrived, in one instance he was promised 20,000 men but received only 6,000 and he was never able to defeat the army of the Catholic Liga.

Even the Danes and Swedes viewed aiding the Germans as the means to gain the end of full English intervention but this was never realised, England provided just enough troops to keep Denmark fighting after their initial reverses and as I mentioned earlier did provide 30,000 troops to the Swedes.

I've sort of run out of steam for now, I can explain Saxon motivations during the run up to and course of the War or answer any follow up questions about the 30 Years War in general if there is any interest.

Sources Europe's Tragedy, Peter Wilson Civil War: The War of the Three Kingdoms, Trevor Royale