Trains from S-u-A go to Marylebone and Euston stations in central London. From Marylebone you could take a taxi to Paddington for Heathrow Connect or Express. From Euston you could make your way to LHR on the Underground with one easy change - no stairs needed - at Green Park station.
There is no easy way to do that journey by rail - the thing you seek to avoid (walking between connections) is unavoidable.
How many in the party? Perhaps check out the cost of a pre booked transfer from S-u-A.
One factor that makes a difference is how many is "we" - the economics of train travel vs taxi are quite different for 2 or 4 people for example.
Do you mean a taxi cab?