I believe actual .22 rimfire military training rifles are pretty scarce. I never saw one in three years in the army. However, I did use a US Army issue .22 bolt action rifle when I was in college in 1964/65. I think it may have been referred to as an M2 Springfield. It was a high quality dedicated target rifle. I don't think there's anything like it today. Fat chance of finding one. The army also used Remington, Stevens, and two Winchester models in addition to the M2. These others are just regular commercial .22 target rifles and none of them very closely approximate any army rifle of that day.
One of the first rifles I bought was another .22 rifle just about the same time I was allowed to use that .22 army rifle. It was a SMLE made in 1914. It was identical to the regular rifle except for the chambering and the fact that the magazine was empty, making it a single shot. But it could shoot, once you got used to the sights, even though it was a little beat up. Again, I suspect it would be hard to find.
I remember one local gun store about 25 years ago had a quantity of Mossberg .22 rifles. I don't know anything about them but they were full-stocked and may have been used by the military. They were bolt action repeaters and had an unusual feature of a plastic finger groove strip set into the grip. I believe that was a distinctive Mossberg feature. The catalog I'm looking at of commercial versions, none of which have full stocks, says it has a clip magazine. Others have a tubular magazine. Anyhow, I can't say if the army ever used them or not. They aren't in any of my manuals. I don't shop for guns anymore, so I can't say how rare they might be but they're probably less expensive than any of the other guns above.
I think the British SA-80 was made in a single shot .22 rimfire version for cadet use but probably in very small numbers.
