IMO there really are at least two questions here:

Whether the British authorities will require a transit visa

--AND--

Whether the airline(s) with which you check in in Mumbai will permit you to check in and board without a transit visa for the UK. It might matter whether VS, Jet Airways, or some airport contract staff are handling the flight departing from India. If you have a RT ticket, there's the same question regarding checking in and boarding the VS (?) flight from JFK to LHR, assuming the same routing in reverse.