I have a different view on this; I think it is quite doable ( we have done a similar trip).
I completely understand the reluctance to try driving on the left. We did it in New Zealand because public transport does not cover all the places we wanted to see. After that trip we swore never again. My sister and her husband drove in Ireland and it almost ruined their marriage ( I am sure you do not want to risk that!). So when we went to the UK together four years ago ( also in May), we agreed to take trains and were very happy with that. I bought the tickets in advance and they were inexpensive. We did not feel we spent too much time on trains or in stations. And we met some nice people along the way. Our itinerary included London, Edinburgh, Oban, Glasgow, a small town in Wales with lots of double LLsin the name, and Cornwall (Port Isaac), with 2-3 nights in each place. We called it the "UK sampler" -and did not feel rushed. We were there for a walking holiday and got in a moderate or a long walk every day, even on travel,days. ( We all have been to London several times before and would be again, so we did not need more than 2nights there).
For you I would suggest this:
After time in London, take the train to Newquay, a journey of 5 hours with one change on the way. This should be your longest train journey.
From Newquay, FLY to Edinburgh. Yes, there is an airport at Newquay, and there are direct flights to Edinburgh (and elsewhere).
From Edinburgh, it is a 1.5 hour journey by direct train (or with one change) to Penrith, gateway to the Lake District. There are frequent buses to Keswick, close by. ( We visited the Lake District on a 2016 trip, staying just out of Keswick in a lovely country house right on Derwentwater (lake). The Cat Bell's were just across the way. The lodge is operated by HF Holidays, if you are interested. They have a three-night minimum and specified start and end dates. They provided the transfer from Penrith station directly to the house.
I haven't forgotten Oxford; it would be your last stop. It is 4.5-5 hours by train from Penrith. (Or more if you do not pick the right train). Spend two nights in Oxford and then take the bus directly to Heathrow for your flight home. Or the train if you prefer; you head toward London and transfer to the Heathrow Connect at some station whose name I do not recall.