You have my total sympathy. Your fear if caused by failure to get numb is 100% logical because after all it is not supposed to hurt.
You probably need a dentist with knowledge of advanced injection techniques. It could be that you have anatomic variation which means that your nerves are in a slightly different place to most people. This is usually the case with lowers rather than uppers. Other causes of numbing problems are here:
Find out why you may not get numb enough, what dentists can do to get you numb, and how to ensure your treatment is painless.
www.dentalfearcentral.org
Since you are in the UK, have you tried the Community Dental Service yet? They do have access to GA and if their dentists can't get you numb (they probably can) then that would then be an option. The trouble is GA is best only for extractions. It is very hard to do complex restorative dentistry under GA.
If you have only tried one dentist re numbing, then all is not lost as failure to get numb is often the dentist's fault (operator error reason). Injecting in the same place 5 times will not work if your nerve is in a different place.
Are you able to go private, then you can handpick a dentist with knowledge of advanced local anaesthetic technique. A poster called bluestringpudding had a similar problem and found a dentist in London who helped her. Mike Gow in Scotland could help you.
It is also a good idea to do an x-ray/CBCT scan to see exactly where your nerves do run. If they can get the LA near enough your nerve, you will go numb. Everyone can be got numb but not all dentists can get all patients numb.
Do you have red hair at all?