As @romaniac and @janemulberry say, there is a time limit for new Bulgarian residents to get themselves a Bulgarian driving license. This is a standard requirement throughout the EU, and, typically, you can't become a long-term resident of a new country and not have a local license. There might be different time periods for those with a non-EU license, and those with an EU license... but I don't know these details. We're not talking 5 minutes after getting your residence permit, maybe 3 months or 6 months or something in that ballpark.
There's a similar requirement with foreign-plated cars. You can't keep your UK (or Germany) registered vehicle here, and expect to drive it for the next 10 years in Bulgaria. On the other hand, if you still have a home in UK/Germany and drive the vehicle back there regularly, you might be able to show that it's more appropriate for you to keep it on the foreign plate.
In many (most) EU countries, I would say there's not much difference between theory and practice. :-) However, Bulgaria appears to be pretty lax about the enforcement of quite a lot of issues, including aspects of driving (speeding, foreign plates, dirty/loud exhausts, foreign driving licenses).
In my experience of traffic stops, they rarely or ever ask for my driving license. They always ask for the car documents, and they want to see a registration document (proof of you own the vehicle), valid insurance, and a current vehicle inspection. The vignette (road tax) has enforcement cameras, so you should have this too, even if you have a foreign-reg car. Even when my car was on its UK plates, they never checked if I'd exceeded the time for re-registering. (However, when I registered at KAT, after a couple of years, they fined me for late registration. Maybe 50 leva.)
I'm guessing they would be a lot stricter if I was drunk, or I'd blown through their speed trap at 200 kph, autobahn-styley. But for random stops or slight speeding, I've had a couple who grudgingly waved me on (after seeing my docs) when they realized I didn't speak Bulgarian (and they didn't speak English).
We have Spanish residence too, and the enforcement there is completely different. The local police can (and do) check whether or not you are a resident, and since what date... and then verify than your car can still be driven on its foreign plate, or that you can still drive on your foreign license. If not, they impound your vehicle, and fine you. They are pretty ruthless. :-)