It's generally pretty hard to get an ICAO Telephony Designator (as is the official name) without holding an AOC. It is possible but you have to come up with a good story why you need one.
When I got mine a few years back the address was:
Civil Aviation Communications Centre
NATS Ltd
Room 203, Control Tower Building
Heathrow Airport
Hounslow TW6 1JJ
Tel +44 (0)20 8745 3420
No idea whether this info is still correct. I'd start with a phone call.