The Commonwealth outlived Kings George V and George VI and, more significantly, outlived most of the former colonies of the British Empire becoming republics. I don't see why it would not survive Queen Elizabeth II as well.
In fact, neither the "wealthy" Commonwealth members (Australia, Canada, etc.) nor the African and Asian members have ever shown any intention to leave or break up the organization.
Being a member of the Commonwealth is totally voluntary so I don't see why a country would remain part of it if there was a strong feeling against it.
Gambia has actually returned this year after having left in 2013.