It's one of those cases where, without knowing the pilot background, one doesn't know whether to write a one-liner, or 2 pages.
In the UK, Class G, you can switch VFR to IFR non-radio i.e. the change can be done entirely inside your brain. This is regardless of whether you have entered IMC or not.
If you are in receipt of a service where the controller cares what you are up to (a radar service, basically) then you should advise them you are changing to IFR due to entering IMC - IMHO.
A flight plan is not needed within UK Class G.