Driving my UK car over to Budapest

I would suggest you look into it more carefully. you would need the car vehicle inspected etc and pay the fees. I myself coming from a rhd car would never suggest driving 1 in Hungary. The safety factor far out weighs the cost factor.
Its not worth the effort or worry I would suggest you look at a good lhd car here or another country that has lhd cars

I agree with most of the previous poster's comments. 

The cost of LHD conversion would be uneconomic although some cars are easier than others.  If your car is Euro 2 or 3 pollution standard you have to pay hefty registration fees. Insurance is not on a par with the UK.  Many things are extras and the excess is high.  It's just not worth it for an older RHD vehicle unless you are an enthusiast or the car is rare or has some other reason to be of high value (like vintage/old timer).   

In any case, you can easily buy an ordinary (Ford, Nissan, Mercedes etc) LHD car with British plates in the UK and drive it about for 6 months and on British insurance.  For import, so long as it has EC Type Approval the paperwork is an annoyance but possible although you might want to use a fixer to take care of the paperwork.

Thank you both. It looks like I will sell it over here in the U.K. in the end !

It is now sold and the flights are booked!

Congrats on the sale, but sad for you that they are gone. :-(
However great to hear all is falling into place and hope you enjoy your time here.
Good luck and a safe trip here..............