In Londons Airspace you will only be refused a weather avoidance turn/climb if that action will put you in real danger from other traffic as opposed to potential danger from the weather. Not much comfort I know.
As you say, UK ATC has no real time weather imformation, and that includes the supervisors who could have tactically delayed your depature through, what appears to have been, an overloaded sector.
Solution? Real time weather radar information for the London FIR available to (at least) the LASs at LACC and Group Supervisors at LTCC and training to use it properly. We've been asking for it for years.
I hope you submitted a MOR or at least a CHIRP on this. Going into writing seriously strengthens our case for improvement in this area. Failing that, try sending the engineering bill to NATS.