If you do Lindesfarne you will need a car from that point. I am sure that you could hire one at Berwick upon Tweed and then head south. But if you bypass that and go straight to Endinburgh I would consider staying a couple of days in the city. You will not need a car there. But when you are ready to leave you could pick one up. Driving on the proper side is no problem for me. I lived there, am left handed, and it makes a lot of sense to me. I have trouble when I come back to the US. Takes me a week or two to get back into US style driving.