Numerous Englishmen served in the various Italian wars during the later 14th century. The Italian states often relied on mercenaries for their land campaigns. That said, the companies themselves would have been comprised of people with a variety of backgrounds. William Caferro wrote a book on the famous English mercenary leader, John Hawkwood. Interestingly, his letters are the earliest example of the form in the English language (thanks Italian archives). His writings reveal a diverse group of people serving together but there were undoubtedly numerous Englishmen.

The Honourable Artillery Company started as a unit that consisted of part time soldiers that started in 1537 (although they had antecedents back to the 11th century). They predated the British Army and were formed from guilds to hone their skills (artillery meaning anything from cross-bows, long bows, through to cannon and guns) to assist with the protection of visiting dignitaries, the defence of London and the country.

As they always fought for the King or the Mayor (except during the civil war when they fought on both sides) they probably can't be thought of as mercenary in the sense that they would serve anyone who paid them but whilst they fought with the army later (and took orders), they were never quite part of it (having a separate royal charter). What is interesting is that they still exist today, providing the the Metropolitan police with part-time policemen ("Special Constabulary") and as the Territorial Army providing reconnaissance as well as fulfilling ceremonial duties in the City of London.