You could do much of that along Sustrans Route 4 in to London and Route 1 out to the coast, 247 miles according to Google
A real coast to coast would be Lands End to Dover, 430 miles if you use Route 3 to Bristol then Route 4 and 1 as above; but having done parts of Route 3, I can tell you it goes nowhere fast.
Having looked at the map the favourite for me would have to be Lands End to Dover along Route 2, a Coast to Coast along the coast..... now there’s an adventure...!
We did our own Coast to Coast this year; however in France not the U.K. Channel to the Med. It was hot but worth every mile of effort.