No, if you are a UK resident then I'm afraid you can't drive a foreign registered car in the UK, full-stop. It's written in other places, but have a look at this DVLA .pdf (Adobe Acrobat) booklet which is downloadable on the link below.https://www.google.com/url?sa=t&rct=j&q ... UHu-ydsfTs
Look particularly at page 3, which is pretty conclusive and says:-
"A UK resident must not drive a vehicle displaying foreign registration number plates in the UK."
As you say your wife is also a UK resident, the rule stands for her too - with no exceptions or periods of grace.
If the car, by your own admission, isn't worth much, then I'd just get shot of it if I were you - but I fully understand that you can get attached to cars.
Having thought a bit more, I'm not even sure if you would be able to register the car in BG in your name before you head off to the UK. I believe that to register a car in BG you need to be a 'legal entity' and as you don't have residency then you have no legal identity in Bulgaria. However, if you still have a company from when you bought any property/properties (pre-2012), then another way of registering it in BG would be to use that company for the vehicle registration. (Perhaps others will correct me here if I am wrong). If you don't have a company, then re-registering it in the UK in your wife's name before you leave the UK could be the other route to take, prior to registering it on a BG number in her name..
One further possible problem I've just noticed. How are you going to validly book a UK MOT when your car has a BG registration? As far as I know, you cannot put foreign cars through a UK MOT test, just like you cannot put a UK car through a mechanical test in BG. There is also no guarantee that DVLA will give you the original number when you re-register it, as they reserve the right to just give you an age-related one.
You really MUST get everything right and it will only take one awkward person, be they a policeman or a border official, to cause you innumerable, possibly insurmountable, problems. I haven't lived there for many years, but I understand that the UK police seize a car in the event of no or incorrect licence/insurance etc etc. What would you do then....?
So many possible problems and I honestly think that trailering to the UK it would be the 2nd most sensible option, with the best option being to sell it onto someone in BG who is willing to risk continuing to drive it around illegally. The risk of doing that may well become greater after BREXIT, of course.