The route I would suggest is London to Edinburgh going up the A1 out of London and switching to the A1M when you get to the north.
Edinburgh to Glasgow.
Glasgow to Liverpool via Carlisle and taking in the Lake Disrict but using the M6 as the main route.
Liverpool to Belfast on the ferry.
Belfast to Dublin
Dublin to Hollyhead on the ferry.
Hollyhead to Cardiff and then take the M4 back to London.
That's a very rough route guide. It's doable but unless you do some very careful micro time management you run the risk of seeing nothing more scenic that the inside of the motorway service stations.
Remember that even hire cars break down on rare occasions. Remember that somebody else's motorway accident can add anything up to six hours to a long journey. Make sure you get a rental agreement that includes unlimited mileage. Remember that currently UK gas prices stand at over $2.00 per litre and that price is never going to go down - only upwards. Buy a good map book.