Jammed Stab - There are no real 'areas of responsibility' in the sense I think you mean because there is no real responsibility - basic service in general, and certainly outside controlled airspace is more of a priviledge than a requirement ! think of it more as 'areas of possibility'.
In response to your request for more info about how to get some external service:
Firstly, consider the airspace definitions as your first place to start. Obviously if you are in controlled airspace ( ie not G ) then you will have someone to provide service. If you are around busy airspace but not inside it, you may find mandatory or recommended radio or transponder requirements, which again will point to who to speak to. ( the oxford/luton comments above are good examples and both excellent at being helpful/providing whatever service they can spare)
Then , if well outside controlled airspace, the LARS coverage map does show coverage, but quite a lot of it is not H24, so that is not always helpful.
Then, you can ask for basic service from anywhere you are flying close to, or over the top of.
Then, there is London/Scottish information, which does have good coverage, but is only really of use if you need info such as METARS or if you are in minor difficulties. They don't have radar service , as stated above. (but they can 'look over the radar operators shoulder')
but to clarify your original question - basic service is indeed really basic. It is really only having someone to talk to. Here is the definition
A Basic Service is an ATS provided for the purpose of giving advice and
information useful for the safe and efficient conduct of flights. This may include
weather information, changes of serviceability of facilities, conditions at
aerodromes, general airspace activity information, and any other information
likely to affect safety. The avoidance of other traffic is solely the pilot’s
responsibility.
Basic Service relies on the pilot avoiding other traffic, unaided by controllers/
FISOs. It is essential that a pilot receiving this ATS remains alert to the fact that,
unlike a Traffic Service and a Deconfliction Service, the provider of a Basic
Service is not required to monitor the flight.
That is from this doc
https://publicapps.caa.co.uk/docs/33...74Issue2_3.pdf
which you may find helpful.
You are, of course, not obliged to use any service or even have a radio when in class G.
Does that help?