is a good account of the development of the cavity magnetron.
The winners generally write history but this seems a pretty unbiased account by three authors from Italy, France and the Netherlands.
Like many other disruptive breakthroughs, the cavity magnetron was the result of a number of related explorations, in technology, in experiment, and in theory. Many other scientists, scattered over quite a few countries, made significant advances. However, exchange of ideas and opinions in scientific forums was getting more and more difficult in the build-up for WW II, as well as between Eastern and Western scientists in the ensuing “Cold War” period. The actual global status of science and technology was known in a rather fragmented way to many engineers and scientists. The military relevance at the time led to teams working on the same type of problem, but in imposed isolation. Although the Birmingham team had brilliant ideas and their share in the development proved decisive, it is fair to say that without the contributions from others, their degree of success and the pace of the progress would not have been so great, or maybe it would have been too late for a timely development of microwave high-power radar in World War II.
British engineers and scientists combined a number of important breakthroughs in making their magnetron which gave the Allies a 2-3 year head start in microwave radar technology. The Tizard mission showed that the Brits were ahead of the US in most areas of radar technology, and the pooled Allied radar work leveraged American expertise in mass production combining the British and American strengths.
The magnetron was invented by an American, improved by a Japanese researcher, and the oxide coatings which GEC added to Randall and Boot's design to improve it came from a French scientist. The Americans commercialised production: 150 different radars derived from it by the end of the war. The MIT Radiation Laboratory textbooks which came from this Allied effort formed the basis of postwar radar and electronics industries all around the world.
The German patent mentioned is a 4 cavity design; we don't know if Randall and Boot knew of it, but even if they did their design used more cavities. Patenting a 4 cavity magnetron design is not the same as coming up with a finished, reproducible industrial product. German scientists and engineers were obviously aware of the magnetron, but mostly preferred the klystron because it gave stable frequencies and was therefore technically 'superior' for many applications - but it couldn't match the power generation capability of the magnetron. Eventually, with pulse-doppler radars needing predictable frequencies, the magnetron was largely supplanted by klystrons and Travelling-Wave Tubes (TWT).
Britain mostly surrendered their technical lead after WW2, though they remained on the cutting edge in anti-ECCM work, and Ferranti (even Marconi, occasionally) consistently delivered good products up to and including the Typhoon's Captor radar, which won against a US/German radar design.