Just to add that if you are undertaking flying training in the UK in a foreign registered aircraft you will also need the permission of the Secretary of State for Transport under Article 225 Air Navigation Order 2009.
True, but only if the instructor is getting paid, and is getting paid for the flight, specifically getting paid for the portion of the flight which is logged as training within UK airspace
Also true, but Article 225 permission easily obtained. Applied to DfT for renewal of our permit by fax at 16.00 on Feb 18 and received it by fax at 09.30 the following day. Nice to be able to compliment a regulator on being efficient!