A Discovery CycleTours van provides round trip transportation between all London airports and Norwich.

FROM LONDON HEATHROW AIRPORT:  The fastest way from London Heathrow (LHR) to Norwich is a 3 leg journey.  First, take the Heathrow Express train from the Heathrow Central Station (clearly marked) to Paddington Station.  Non-stop trains leave every 15 minutes and take 15 minutes to reach Paddington Station; then, take the Underground (the Tube) Hammersmith & City line east to Liverpool Street Station (9 stops, approximately 20 minutes).  From Liverpool Street, trains to Norwich leave every 30 minutes for the journey to Norwich, arriving in 2 hours. 
Alternately, you can take the Underground (Tube) all the way to the Liverpool Street Train station.  Take the tube from Heathrow to the Holborn stop, and get on the Central Line east to Liverpool Street, 4 stops. The tube will take about one hour to get you from Heathrow to Liverpool Street.

FROM LONDON GATWICK AIRPORT:  London’s Gatwick airport (LGW) has its own train station located in the South Terminal, linked via a free transit train to the North Terminal.  The journey to Norwich is a 3 leg journey.  First, take the train from LGW to Farringdon Station, a 40 minute journey.  Then, transfer to the Underground at Farringdon Station and go 3 stops east to Liverpool Street.  From the Liverpool Street Station, trains leave every 30 minutes for the 2 hour journey to Norwich.

FROM LONDON STANSTED AIRPORT:  Stansted airport has an express train (The Stansted Express) traveling from the airport train station to the Liverpool Street train station.  Trains leave Stansted every 15 minutes and arrive at Liverpool Street in about 45 minutes. From the Liverpool Street Station, trains leave every 30 minutes for the 2 hour journey to Norwich.