What are reasons that different trainee pilots put on on versus the another?
Requirements and suitability.
Take the RN. They fly Lynx and Sea King, not Apache. So a RN student will not get to fly the Apache. If, at the time, the RN are short of Sea King pilots, then most of the students will go to Sea King, and vice versa.
Of the three, the RAF only fly the Sea King, so RAF students don't get to fly the other two.
The Army fly the Lynx and the Apache. Again if more of one sort are required, that will be where students go, in general. Although they all fly the same in terms of using the cyclic/collective/pedals, the Apache is very demanding to use in it's role as an attack helicopter.
I was a Lynx QHI, but always felt that the Apache would have been beyond my intellectual capabilities.