Although the Tube map pretends they don't exist, Finsbury Park has direct services to both Moorgate and Farringdon, both giving connections to Crossrail. In the short term a change at Paddington will be needed for Heathrow, but from the end of next year (if all goes to plan) there will be direct trains from both Farringdon and Moorgate to Heathrow.
Finsbury Park to Heathrow by Piccadilly Line is 65 minutes, via Farringdon will be about 45.
And of course it's not just about Heathrow - connections to East London and Docklands from north London will be much faster too.