National Express operate bus and coach services between most UK cities. They also operate the European coach network known as Eurolines.
The National Express Flightlink service operates from London's Heathrow and Gatwick (and many other) airports to Bristol. Services from Heathrow airport depart from the Central Bus Station. The journey time is about 2 to 2.5 hours. Services from Gatwick depart from both terminals, and the journey time is 3 to 3.5 hours.
It is advisable to book your tickets in advance where possible. All of the links above contain on-line booking facilities.
All of these bus and coach services will terminate at Bristol Bus Station, Marlborough Street, Bristol. This is no more than 3/4 of a mile (1.2 km) from the University.