I could do Paddock Wood in Kent to Edinburgh in about 9 hours in my truck (at 56mph max). It depends how fast you drive. Best way is straight up the A1/A1(M). If you want to go a bit more scenic, and take about the same time, when you have crossed the Tyne at Newcastle look out for about the 4th turn off , which is the A696. This becomes the A68 and takes you over Carter Bar and on to Edinburgh.
Dont know from London but you can drive from Birmingham to Edinburgh in about 6 hours. Have you considered flying though you can get some really cheap flights only have to check in an hour before and it only takes about 1hr 10 mins in the air.