These are the countries that were declared rabies-free: Several countries in Europe have been designated rabies-free jurisdictions: Austria, United Kingdom, Ireland, Belgium, the Netherlands, Luxembourg, France, Switzerland, Portugal, Italy, Spain, Greece, Malta, Germany, Denmark, Norway, Sweden, Finland, Latvia, Estonia, Czech Republic, and Republic of Cyprus Iceland. (source Wikipedia).
In all countries of Europe, any suspected case of rabies is dealt with in a very swift manner. The normal procedure would be: if you get attacked by a dog with a known owner, the owner needs to provide proof of a current rabies vaccination. Any suspected case of rabies (bitten by a wild animal or an animal where no proof of vaccination can be provided) is referred to a hospital, the person is isolated and undergoes a course of anti-rabies treatment (which from what I've heard is extremely unpleasant).
Overall, even in areas with a high population of wildlife it is highly improbable to be exposed to rabies, especially as there was lots of effort to eradicate the disease through vaccination, dropping the rabies vaccine bait for foxes etc.
Most of it will get sorted out with the Brexit anyway and you'd be facing a higher probability of eating a chlorine washed chicken than getting a dog from "dodgy" parts of the world.