Is the popular mythology wrong to assume that the British invented Radar first
Wikipedia on Robert Watson-Watt:
His early work was using radio to detect thunderstorms
In January 1935 ... he also mentioned in the same report a suggestion that was originally made to him by Wilkins that radio waves may be capable of detecting aircraft.
On 12 February 1935, Watson-Watt sent the secret memo of the proposed system to the Air Ministry, Detection and location of aircraft by radio methods. Although not as exciting as a death-ray, the concept clearly had potential but the Air Ministry, before giving funding, asked for a demonstration proving that radio waves could be reflected by an aircraft. This was ready by 26 February and consisted of two receiving antennas located about ten km away from one of the BBC's shortwave broadcast stations at Daventry. The two antennas were phased such that signals travelling directly from the station cancelled themselves out, but signals arriving from other angles were admitted, thereby deflecting the trace on a CRT indicator (passive radar). Such was the secrecy of this test that only three people witnessed it: Watson-Watt, his colleague Arnold Wilkins, and a single member of the committee, A.P. Rowe. The demonstration was a success; on several occasions a clear signal was seen from a Handley Page Heyford bomber being flown around the site. Most importantly, the prime minister, Stanley Baldwin, was kept quietly informed of radar's progress. On 2 April 1935, Watson-Watt received a patent on a radio device for detecting and locating an aircraft.
In mid-May 1935, Wilkins left the Radio Research Station with a small party, including Edward George Bowen, to start further research at Orford Ness, an isolated peninsula on the coast of the North Sea. By June they were detecting aircraft at 27 km, which was enough for scientists and engineers to stop all work on competing sound-based detection systems. By the end of the year the range was up to 100 km, at which point plans were made in December to set up five stations covering the approaches to London.
So, very rapid (only a few months) progress in implementing a system once there was a desire for and funding for it.
Radar Development in Germany for Aircraft Warning
GEMA built the first radar transmitter in the autumn of 1934 for detecting ships. The radar operated on 50 cm wave length and could find ships up to 10 km away. In the summer of 1935, a pulse radar was developed with which they could spot the ship the "Koenigsberg" 8 km away. This radar unit used the "Braunschen" tubes and had an accuracy of 50 m. A magnetron had been tried but the frequency was not stable, and as such, tubes were used. A wave length of 60 - 80 cm was used so the sender and receiver could be spaced close together. An airplane at a height of 500 m and a distance of 28 km could be seen. By 1935, they had built the first successful radar unit.
It also says,
Huelsmeyer had invented and demonstrated his radar in 1904. However, his invention was an idea too early for its time.
I don't see any claim that British radar wasn't vital, so I won't answer that.
And Watson-Watt did invent /a/ radar device, and got a patent on it. However, clearly what he invented was a particular implementation of a radar device (and not, the concept of radar, nor even the first practical implementation of it).