I think that Mini and JL have, between them, got this right, for the first UK airline to operate into the USA was indeed Imperial Airways on a service from Bermuda. It didn't come all the way from London though.
There was just one aircraft, Short S.23 Empire class G-ADUU "Cavalier" 4-engined flying boat, taken from the fleet which ran to Asia and Africa, which was shipped out in crates to Bermuda and put together there. Crews were on long-term detachment. First scheduled service was 17 May 1937 to New York (complete answer to question), it took 5.5 hours. British Airways ran a special Concorde flight on the route 50 years later to commemorate. It took a bit less time ! The US terminal was at Port Washington, on Long Island, initially approached over Manhattan for air-to-air publicity shots from another aircraft with a New York City backdrop. Meanwhile, on the same day, Pan Am started on the same route with an equivalent Sikorsky S.42 flying boat. Both airlines published a joint timetable.
In winter 1937-8 the service was operated to Baltimore instead, being cautious about icing conditions any further north. The next winter they didn't do so, and alas Cavalier was lost mid-trip after departing New York on 21 January 1939 due to icing of engines and airframe. Most on board were rescued after many hours in the water. It appears the aircraft did not have life rafts. There wasn't another Imperial aircraft to hand, so after nearly two years that was the end of the first UK airline schedule.
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