In 1936, who was the first person to fly across the Atlantic in nonstop from east to west, proving that an airline service between London and New York was possible?

The first non-stop transatlantic flight from east to west seems to be in 1919 by a British airship R34, actually part of a return journey carrying about 30 people. They left from East Fortune, Scotland and arrived in Long Island, USA after 108 hours of flight.

The first non-stop solo flight in an airplane was by Jim Mollison in 1932, flying from Portmarnock, Ireland to Pennfield, New Brunswick, Canada.

The flight to which you refer is presumably that of Beryl Markham who became the first woman to achieve the feat in 1936. She flew from Abingdon, England to Cape Breton Island, Nova Scotia, Canada.

https://en.wikipedia.org/wiki/Transatlantic_flight