Heathrow to Kings Cross

Yes, Piccadilly line all the way.
But, if you are pre-buying tickets from King's Cross (and that is a lot cheaper), don't forget to allow plenty of time for all the delays at Heathrow before you even get on the train: taxiing, disembarking, walking through endless corridors and above all waiting at immigration. Plus the risk your flight might be late.

Don't look for Eastcoast trains anymore - they have yet again gone back to the government to run, the last company had to give the keys back. The most recent failed group called it Virgin Trains East Coast.

The new name and paint job is LNER, London Northeastern Railway, known as LNER.

Same staff, same trains, new look.

one definition of madness is flying to LHR then taking a train to Edinburgh

say 1 hour to get through LHR

1 hour LHR to KIngs Cross

4.5 hrs Kings Cross to Edinburgh

An equal definition of madness is travelling from central London to Edinburgh and NOT taking the train.
chouston, have you investigated a connecting flight from Heathrow? Or, even better, a flight from your starting airport direct to Edinburgh?

Easiest? The Piccadilly line

Fastest? Perhaps that depends upon the day and the time