Are you going to have a tube pass or buying a single ticket ? You say "our" which means more than 1, is it 2/3/4 or you ? In this case, if you are buying single tix for multiple people, at that time of morning, a taxi over to Paddington might be similar in price and be less stressful with luggage, etc.

Of course, if you want cheapest, just take tube via green park all the way to heathrow, probably will take an hour or so - just be aware of any clousres as mentioned.

Looked up schedule:

Time Details Information
05:34 Finchley Road Underground Station
05:47 Green Park Underground Station

05:55 Green Park Underground Station
06:41 Heathrow Terminal 4 Underground Station

Plenty of time if you want cheap.

Cheers.