I can’t prove one way or another Queen Elizabeth shot pheasant or not, but she certainly shot a stag at age 19. She had no objection to pheasant shoots, however:
https://abcnews.go.com/amp/Internati...d=82100&page=1
If you ever get a chance, do visit Sandringham. Very impressive! At the time I went there, there were pheasants outside the main gate, and this was in June. I got a behind the scenes tour from the kennel man, Bill Meldrum, at the time.
https://en.m.wikipedia.org/wiki/Sandringham_House