London to Bristol is 190km or 118 miles, about 2 hours drive by car. Which in the big scope is not that bad...unfortunaty I don't think your driving, hitching this portion of your trip might be an option.
You can take a bus from
Victoria St. Stn. in Central London. looks like National Express has buses leaving, 2 every hour for the rush hour, and one every hour after that. The travel time is about 3 hours and you will be dropped off at
BRISTOL, Marlborough Street bus station, Avon
Hope that helps ya some, oh and the cost was about 10£ ( I think if you book online, you might save a few pounds
National Express UK. )
And Metalsign,