I did a similar thing in reverse and loved seeing the 'real' USA.
I became vert friendly with a group on the net and stayed with different ones of them started in VA - very like the UK in many ways, went to NC which was like being in a sauna, then to ID which I absolutely loved then on to CO which was also beautiful.
I didn't drive I was driven by these people!
So, I would suggest that you either start or finish in London as that is where you will fly to and from. Whatever you do I would say do not even attempt to drive in London - I know the city fairly well but never drive there. It is easy to get around with the trains and buses.
UK roads are nothing like those in the US. Even the motorways (freeways) aren't straight and, as said, it can take far longer to travel any distance because of the amount of traffic and hold ups on the roads.
Gas prices are astronomical, (gas $7.8 pr
US gallon) (Diesel $8.13) Trains are easier but also costly.
Traffic lights are different in that they are rarely overhead, but by the side of the road and we have roundabouts and I never saw one of them in the US!
Dorset is a beautiful area - as is Yorkshire. A rough distance of 250 miles but without hold ups will take over 4 hours to get to.
I will be living on the Isle of Wight and would be more than willing to have you stay and take you around there and possibly Dorset too (I am living in Dorset at the moment) so places to stay are easy to find (with friends)
The West Country ( Dorset, Devon, Somerset, Cornwall) are all beautiful. There is such diversity of landscapes.
Make friends with people on forums and then you will find cheap lodgings!
PM me of you are interested.