Let me answer some of the queries raised by respondents:
Car insurance - It's true that most companies won't deal with non-residents, but there are a couple of intermediaries that can arrange cover:
Camper Van Insurance and Motor Vehicle Insurance
https://www.hertsinsurance.com/walkabout.php
Any UK cover is automatically valid in most countries in Western and Central Europe, but only for third-party (public liability) risks, unless the cover specifies comprehensive.
Car tax - called Vehicle Excise Licence or Tax Disc, can be paid when you register a car in your name. Easier way to deal with it is to buy a car that still has several months of tax left, or ask the dealer or existing owner to extend the tax, at your expense.
Ther is also MOT - a kind of safety and environmental test - required annually by all cars three years or older. Again thing to do is to only buy a car that already comes with long MOT, or get it MOTed before parting with your money. Honest sellers shouldn't mind, but dishonest ones will, as they know it won't pass. Never touch a car without recent MOT.
Registration. When you buy a used car, the seller will complete part of the registration certificate and send it to effect change of registration. You get another part for your record. Within 2-3 weeks you should get a new registration document in your name - you need a UK residential address for it, but it can be a friend's or relative's. You cannot give a business address or a hotel's.
Right-hand and left-hand drive issue. Whichever you get, it will be 'wrong' either in UK or continental Europe. Most experienced drivers quickly get used to driving on the other side. It takes me about 20 min of concentration when I cross the Channel with my car. Having a good navigator on your passenger seat will help, as they can tell you when it's safe to pull out to overtake (pass) when the steering is on the 'other' side. If you prefer, you can get LHD models in UK.
Reselling the car after your trip. This can be time-consuming and frustrating. First contact the dealer you got it from - they may take it back (at a loss to you of course). You can always put it into a car auction. Most big cities have at least a weekly auction. Just take it to an auctioneer with all your papers, driving licence and passport. If desperate, just take it around used car dealers and ask if they would make an offer. You won't get much and you'll be offered bottom prices, but it will be quick. You will of course get more for it by selling privately, but it can take a long time to advertise (often online), wait around for phone calls and viewings and finalise a deal (there are a lot of time-wasters). You can leave the car with a friend or relative to sell it for you and send the money to you afterwards, or if it's old and decrepit, just scrap it - you may get ÂŁ50 for it.