As an ex-FIR (London Information) controller, I would say call 'Information' to request a joining or crossing clearance.
London Information has a direct link into the Airways computer so can;
Firstly, see if your details are filed (in the LATCC) database.
If not they can take your details on, hopefully a quite frequency, and then input them direct to the LATCC computer.
Secondly, contact the sector rapidly for the joining clearance. If the sector won't answer the phone they can 'run over' and present it to the sector in person. That is something that the majority of APC units cannot do !
The FIR controller can also take an airborne flight plan should you require it. But please make sure that you have made it out in advance, so that you can just read it to him/her. There is nothing worse than someone trying to give you details from memory !!
Cheers
PS I am only an ex-FIR chappie as the let me do Radar now...