The problem here is that the present National Anthem is, like it or not, the anthem for the whole of the UK. LOH&G has the advantage that the country name is not mentioned and therefore could be substituted for GSTQ/K if the UK ever became a republic (in fact, the UK is already in political terms, a republic - it just happens to have an hereditary president - but that's an argument for another day).
If, however, devolution becomes complete, then there is a case for Jerusalem to become England's national anthem (although I am rather fond of He is an Englishman from HMS Pinafore as an alternative).