I and many others think God Save The Queen has a fairly boring and uninspiring tune (the lyrics are not exactly inspiring either, IMO).

For a British (as opposed to English) anthem, it would be better to have a rousing tune such as Rule Britannia or Land of Hope & Glory, but as others have said, the lyrics may not be as appropriate today as they once were.

For England, I don't know any existing song that would really fit the bill, having a good tune and lyrics appropriate to today. It would be better to write a new anthem. Maybe we could do it here as a joint effort, as per Jim Lad's songwriting by committee attempt.