There isn't really a market for European LHD cars in the UK (there is however a niche market for American cars though). I assume you'll have a basic European, and EU registered LHD car (otherwise some of the following might not apply to you).
Most dealerships won't buy your car, as they won't be able to resell it:
- They need to properly import it. This means they need to modify it to pass MOT, and also lots of paperwork, which costs money.
- They won't be able to sell it, as there isn't really a market for European LHD cars.
- Also their insurance usually only covers RHD cars anyway.
What you can do is:
- Try to advertise it online. There might be someone willing to buy it (usually for parts though)
- Go to a scrap yard and sell it as parts, as most of the parts in an LHD car is still usable for an RHD car. This is usually an option if your car is common enough, so it has proper market for parts (sometimes they actually ship the parts back to mainland Europe - I know a lot of used Ford parts in Hungary is actually from scrap UK cars).
- Sell it in your home country, and use the money to buy an RHD car in the UK.
- Just register it with the DVLA, and use it in the UK. Usually the only change you need is to modify the headlights to beam to the left instead of right, and it will pass MOT (given it is in good condition and would pass MOT on its own). You might also want to buy a speedometer that measures speed in miles as well, but that's not mandatory (unless you want to use it as a learners car for someone else - Yes, you don't need to have an RHD to pass the driving test, but it needs to measure the speed in mph). Note that it's not that hard to drive an LHD in the UK (personally I got used to it really quick - I have more issues driving RHD cars, as the gear shift and the rear mirror is on the other side)
Note that you can only drive your non-UK car in the UK as long as you are considered a non-UK resident. Once you settle down you will be considered a UK resident and you can only drive your non-UK car until the border, and you have to tow it from there (though most of the time no one will care until you get a UK driving licence).