The Japanese, Italians and Swedes all made DB601 and DB605 engines under license. The Germans supposedly were considering the manufacture of some Italian aircraft. It was the people within the Nazi party who did not get along.
Merlins got manufactured in the USA because the British were trying to move production out of range of German bombers. As German bombers ceased effectively bombing British production, this all got less critical. Of course, every bit of production helped. It helped that the Merlin was a good engine that met all sorts of requirements. The Spitfire XVI puzzles me. Why not switch production over to Griffon engines? I suppose that there was a determination to not shut down production lines.
I have Alexander de Seversky's Victor Through Air Power here, and he claimed that the USA was stupid for making Merlins, rather than Napier Sabres. He would not have been aware of the Sabre's lack of reliability. The Sabre was not a good substitute for any engines that the Americans were using, particularly the Pratt and Whitney R2800.
The British installed R2800s in most of their Vickers Warwicks. The Warwick was big twin engined aircraft, and the Sabres and Centauruses simply were not available. I wonder how an R2800 would have worked on a Hawker Tornado or Tempest? The Fairey Barracuda was too big for its Merlin engine. Wright R2600 anyone? How about four R2800s on the Short Stirlings?
The Americans had difficulty manufacturing Hispano cannons.
Most early American turbojets were copies of British designs.
The British did much of the debugging on the Vought Corsairs for operations off of carriers.
The American tweaked some Spitfire IXs to increase range. I don't think they were serious. Without drop tanks, a P47 was not longer ranged than a Spitfire, but the Americans aggressively developed drop tanks. The British regarded the Spitfire primarily as an interceptor, not requiring range. When they wanted to escort their bombers, they used Mustangs. The Royal Navy installed P40 drop tanks on their Seafires out in the Pacific.
A team from North American visited Supermarine to get tips on the design of lightweight aircraft, which led to the P51H. Did Supermarine get tips on how to make things manufacturable?
Did the British manufacture any American designs?