As an aside, does the British Government provide health care to it's veterans?

There are two answers to this. The veterans are UK citizens. The UK provides health care to all its citizens, free at the point of use. So yes.

There is, however, continuining arguments about whether veterans need special care over and above the 'normal' NHS. An example would be do we need specialist hospitals which concentrate on battle-related wounds, PTSD, and so on. We used to have them, but we have less now [I think].