The Brits leapfrogged the Germans (and everybody else) by inventing the multicavity magnetron. This made microwave radar possible. German sets were still UHF and VHF (Chain Home was ridiculously long wave HF. Yet it was just good enough.) While the Germans and Japanese eventually produced limited number of microwave sets, they never produced them in the numbers that the Allies did.
When the Tizard commission carrying the latest British scientific inventions arrived in the U.S. by ship, they landed first in Boston, where they were greeted by Roosevelt’s scientific advisor Vannevar Bush. (No relation to Prescott or the George’s.) Bush, just happened to be a founding director of Raytheon Company. He showed the multicavity magnetron to Percy Spencer, later famous as the inventor of the microwave oven (in 1947!). Although the British scientist who carried the tube over from England had kept it in a case chained to his arm, Spencer took it home to play with. The next day he proposed that instead of machining it out of a solid piece of copper, he could produce them by stamping sheets of copper, bolting them together and then machining out the inner surface of the resonant cavity. In the event Raytheon went on to produce over half of all the magnetrons made during World War II.
Airborne surface search radar turned the tide in the Battle of the Atlantic. In this role, cheap microwave radar was a truly a war winning invention. The U.S. spent more money developing radar during the war than it did on the Manhattan Project. It has truly been said, that while the atomic bomb ended the War, radar won it.
Plus the breaking of the German and Japanese ciphers.