Cost: BA returns from Heathrow to Newcastle start at £69. Journey time is timetabled at 1h 10m. Newcastle Airport is 20 minutes by cab or 25 minutes by Metro from the city centre. Heathrow is further, and adds £14 to your journey cost if you want to take the 15-minute Heathrow Express.
It's possible to get Standard class train returns from £30, First Class from £59 but these are heavy advance purchase and limited to the time of day you can travel. The fastest timetabled journey from London-Newcastle is 2h 40m, most average about 3h 10m. Both stations are in the city centre.
This is probably the only train route in Britain that effectively competes with quality domestic air travel.