Flight
Bristol Airport (BRS) is the closer airport, however, Heathrow is easier to fly into.
Train
The app "Trainline" is a great tool that we recommend using if you plan to travel by train. Rome2Rio is another great resource. Travel by train should take between 2.5 - 3 hours.
Travel Note
Travel from Heathrow to Bristol is available through Megabus. There are up to 6 buses per day, leaving from Terminals 2 and 3. The bus will drop you off at Bristol Meads Railway Station, where you can get a taxi to Blagdon. If you are staying in Bristol, the station is about a 20 minute walk to town, or you can take a taxi into town.
Travel Note
Heathrow Airport has a plethora of rental car companies that you can use. Please make sure you look into the required documents to rent a car. Don't forget to drive on the left side of the road!