As a follow up question, on the return trip to Heathrow on Sunday, I'm considering taking the tube back but I can't seem to find what time the Piccadilly line starts to run on Sunday at the Transport for London page. Maybe I'm just missing it. Would it be running by 5 AM?
As others have said, the Picadilly line has a Night Tube service, all night, on Friday and Saturday nights (i.e. into Sunday morning) - info here:
Transport for London - The Night Tube
Worth noting the Piccadilly line's Night Tube service does not run to Terminal 4, nor does it operate on the Uxbridge branch (which heads off to the north west London suburbs, so is irrelevant if coming from central London).