There is a whole long list of other nations from which British regiments were raised from French to Spanish to Chinese and Swiss regiments were being formed for the Crimean War. Not to mention individuals joining normal regiments. One of the best known being the Swiss chappie in the 24th at Rorkes Drift. The Royal Navy went on a warm and upright basis traditionally for recruiting sailors.During the Napoleonic Wars the British recruited troops initially from the occupied Electorate of a Hanover and then other German states as the King's German Legion, they fought overseas and were based on the Isle of Wight as due to legal restrictions they couldn't be based on the British mainland. A similar limitation using say the Philippines and Puerto Rico if after 1898 could help assuage some of their concerns, especially being able to pay them less. Even then it might necessitate a trade off though.
It is said that Napoleon's father wanted him to join the Royal Navy as a Midshipman. George Washington was also said to want to a career in the Royal Navy. One could have had a young Midshipman Bonaparte serving under the elderly Captain Washington in the service of the Crown in an AHTL. The over riding principle was that, if you will swear allegiance to the Crown, we don't care where you came from. The German Regiments referred to were different. They were in the armies of their own states, albeit under British command. Not unlike the foreign units of governments in exile in WW2. Thus actually being units of a foreign army.
It would take an American to comment upon what means would be politically plausible to recruit foreigners abroad into the US Army other than immigrants who have gained US Citizenship or drafted in time of war.