Yes, you need to apply to transfer at least twelve weeks before your TX date and everything transfers across. The tx medical constitutes an entry medical and your fitness test carries forward. There is a JPA form for it but I don't recall offhand what it is.
Get in touch with the recruiting officer at your nearest unit to discuss branch or specialisation.
Note, the above assumes that you've reached the trained strength and that you've finished your return of service. If you're being discharged SHORE or SNLR then it's unlikely that you'd be permitted to join the RNR, particularly as you appear to already have been a re-entrant.
Empirical evidence suggests that the ease with which people transfer across very depends on whose desk it lands on in 'the 'MoD', or wherever these things seem to disappear off to. Some people seem to go across with ease, while others have their paperwork go into a black hole.
Ref the JPA form, I understand from a recent transferee that only applies to transfers between the regular forces.