Definitely has to be a licenced aerodrome I'm afraid: ANO Part X article 101 says:
PART X AIR TRAFFIC SERVICE EQUIPMENT, AERODROMES, AERONAUTICAL LIGHTS AND DANGEROUS LIGHTS
Aerodromes – public transport of passengers and instruction in flying
101 (1) An aircraft to which this paragraph applies shall not take off or land at a place in the United Kingdom other than:
(a) an aerodrome licensed under this Order for the take-off and landing of such aircraft; or
(b) a Government aerodrome, or an aerodrome owned or managed by the CAA, notified as available for the take-off and landing of such aircraft, or in respect ofwhich the person in charge of the aerodrome has given his permission for the particular aircraft to take off or land as the case may be;
and in accordance with any conditions subject to which the aerodrome may havebeen licensed or notified, or subject to which such permission may have been given.
(2) Subject to paragraph (3), paragraph (1) applies to:
(a) aeroplanes of which the maximum total weight authorised exceeds 2730 kg and which are flying:
(i) for the purpose of the public transport of passengers;
(ii) for the purpose of instruction in flying given to any person for the purpose
of becoming qualified for the grant of a pilot’s licence or the inclusion of an
aircraft rating, a night rating or a night qualification in a licence; or
(iii) for the purpose of carrying out flying tests in respect of the grant of a pilot’s licence or the inclusion of an aircraft rating or a night rating in a licence;
However, you can train on a Permit aircraft, with certain conditions (ask the LAA). C of A categories don't exist for EASA-controlled aircraft anymore, so the public transport thing doesn't apply any longer.