I think it would be fair to describe many of the bands of fighters that associated with Arab revolt as mercenaries, although these were obviously not organized like private military companies.

There is another issue of "what is a mercenary?" Under this definition of a mercenary here the issue is whether the tribesmen counted as "a national of a Party to the conflict [or] a resident of territory controlled by a Party to the conflict." Of course by this definition private military companies aren't really mercenaries either so I think we can probably ignore this aspect as it is commonly understood.

Any of the Bedouin from the Nejd would therefore count as mercenaries in a strict geneva convention definition. I would make a broader case that most of the bedouin tribes not directly loyal to the Hashemites who nonetheless sided with the Arab army were often mercenary in a colloquial if not strictly legal sense as well.

Cossacks fought for at least one side in each World War. Whether or not Cossacks are, well, anything is difficult to answer, though, so I'll say "Maybe," until a flaired user comes along.