You need to be very careful about this plan. If you have never travelled outside London you might have this vision of broad empty roads and free flowing traffic. The truth will be painfully different.
Work on the basis of averaging 40 miles per hour in the car. You should do better but jams, delays, getting lost, breaks and slow country roads will slow you massively.
Don't underestimate petrol costs as well which will be substantial given UK prices.
I am almost tempted to tell you to get the train from London to Bristol, hire a car to see the south west, drop it back in Bristol then fly to Scotland. Hire a car, drop it back in Scotland. Take the train to York for a couple of days then get another train back to London.