Like someone said, Heathrow Express to Paddington and then Paddington to Gloucester by train. No stairs if you do it that way.. 6 suitcases will be a bear though. Can you narrow down how much luggage you bring? I've done the tube by myself with 3 large bags but oh man did it suck.
I agree with this. Obviously, travelling with a child and six suitcases is a challenge however you do it. But Heathrow Express to Paddington and Paddington to Gloucester by train is a very good option, although relatively expensive.
The Heathrow Express trains are designed for people with lots of luggage, and are far easier than trying to take the tube. The interchange at Paddington should be pretty easy, especially if you already have your tickets in hand.
No way would I choose to drive.