It would probably take the best part of 2 years from you kicking off your application to getting trade qualified, by the time all the various checks etc have been done so probably not worth it from either yours or the TA's perspective if youre only going to be around for 2 years.
Having said that Im not a recruiter, and if you dont ask you dont get. For a definite answer try contacting whatever unit youre thinking of joining and asking directly.
You are ineligible whilst only in possession of a working holiday visa. You need to have been granted indefinite leave to remain in the UK in order to enlist. Ideally you should also have 5 years residency within the UK, though in exceptional cases this can be waived.