Historically all recruits were instantly a private man.
But if you joined a Guards regiment you were a Guardsman, if you joined a fusilier regiment you were a fusilier.
The only regiment which might have used Kingsman officially, though I doubt any used it in practice, is perhaps the 4th Kings Own but even that I'm doubtful. It is used today by the Duke of Lancaster's regiment and is equivalent like all others to private.
NW regiments just have recruit,private,regular,fusilier,guardsmen and all those others because it helps to create a chain of command.
Historically a British company at it's full strength of 100 men, though often the real figure was much less. Would have something like 3 officers,2 sergeants,4 corporals and the rest privates. I don't know the musician complement though.