There are two issues here
a. Training on a private Cof A aeroplane
b. Payment for flying training.
If there is no payment, the flight is private in all respects and you can train on any aeroplane.
JAR-FCL requires training aircraft to have a certificate of airworthiness accepted by a JAA State. This is not law and depending on the interpretation of "certificate" could include Permit aircraft.
If payment is made then the AIC White 49 comes into play.
If you change ownership, then you must register this with the CAA.
Before an instructor can teach you he must register as a Training Facility with the CAA giving details of your aircraft and the licensed aerodrome where training will take place.
If the aircraft is group owned, your problem is finding an instructor who will work for nothing so you can have a cheap PPL!
Bit like finding someone to take you out to lunch every Sunday!