Shortest route is A1/A14/M11.
M6 could be faster IF you don't get stuck in traffic (M6/M6TOLL/A14/M11) and depending on your motorway cruising speed.
My route from Edinburgh would be:
A703 to Peebles,
A72 to the A707 split (signed Selkirk)
Cut through Selkirk back onto the A7 to Hawick
Then pick up the A6088 Boncester Bridge road across to the A68 at the border.
A68 south to carter bar, keep straight onto the A696 to the A1 at Newcastle.
[Also nice is to keep on the A68 all the way to and across the A69 and follow the A68 to the A1 just at Bishop Auckland.]
Then follow the A1 for 200 miles ish.
I hate the A14/M11 bit - the road has miles of specs style cameras on it which causes miles of slow traffic - I would go almost to M25 on A1 then cut across A602/A414 around Ware and Harlow.
This route may take you longer but is more fun

Also on the way home you can drive it backwards with all the fun bits at the end!
Cheers,
Robin