I have been waiting over 5 weeks.
Hi B'dog. Welcome to PPRuNe, and to the real world of dealing with our lovely CAA.
Although the actual work to issue a Licence can easily be done in an hour, I usually advise people to wait at least 4 weeks before even enquiring with the CAA, and to expect it to take around 6 weeks.
As others have said here, complaining to, or about the CAA is a complete waste of time, as they have no effective oversight, and so they don't need to, nor do they, give a toss how bad their service is.
MJ
Ps. When your Licence arrives, don't forget to sign it, and take it straight to the CFI of your Flying School and have him/her check that they have printed it correctly!