Before anyone goes down the wrong track;
JAR-FCL doe not require a licensed aerodrome for PPL training. It merely requires certain standards are met.
The UK decided that it's way of ensuring that the minimum requirements are met is by requiring training to be completed at a licensed aerodrome.
As for what can be done at an unlicensed aerodrome;
In terms of CS-23 or CS-VLA aircraft - Anything where the pilot under training holds a valid license.
However, that statement is limited by the fact that much training such as Multi and IR etc requires an FTO approval..........and (as far as I am aware currently) you will not get it at an unlicensed aerodrome.
Regards,
DFC