London > Edinburgh. Which way to go?

Take the train. London - York - Edinburgh or west via Liverpool.

With only two days, whether you drive or go by train the Western route is going to be a bit of a whistlestop tour. By either mode Eastern Wales is a bit of a deviation. By either coast train is going to be a lot easier and faster than driving with only two days in hand.

York especially, but also Durham to a fair degree are not car friendly cities.

What attracts you specifically to Liverpool?

I know Chester isn't Wales, but my suggestion on Day one would be to get an early train to Chester- journey time just over two hours (driving would take at least 4 hours) then some time in the early afternoon take a train every 15 minutes to Liverpool (journey time around 45 minutes). Stay overnight in Liverpool. Do whatever you want to do in Liverpool that first day.
Chester is another city which struggles with cars and encourages park and ride.

Really an extra day for Liverpool would be very beneficial, as Chester would be better as a full day.

Then next day Liverpool to Preston on the train, change for Oxenholme then take a bus through the Lake District stopping where you want to on the way to Keswick, Then bus to Penrith every 30 minutes for an evening train to Edinburgh (journey time under two hours, quite a bit less than driving). A Day 4 (to give two days in the Lake District would give a better experience)

On the East Coast an early train to York- day and night there. next day early train to Durham, spend the morning there, early afternoon train to Newcastle (what do you want to do there?), then evening train to Edinburgh. Again an extra day (minimum) would be very useful.

A Day 4 would give time for Hadrian's Wall and/or Northumberland.

Personally, I would take the train up the east coast mainline and break up the journey at York- there's plenty there to fill a couple of days (York Minster, Jorvik Viking Centre, National Railway Museum, etc.).