What Britain actually kept secret about radar

How would the Pacific War change if Britain withheld the cavity magnetron from the United States?

The effect would probably be delay rather than permanent denial, because magnetron work was not uniquely British; particular cathodes, applications, and implementations were the more sensitive knowledge.

Withholding the cavity magnetron would not stop American radar development altogether. Other countries were already pursuing related magnetron work, so secrecy would more likely delay particular systems. The oxide cathode and the practical applications built around the device as more closely held advantages. A technology's strategic value often lies in manufacturing details and implementation rather than in one headline invention.