You thought that the British monarch was automatically the Head of the Commonwealth, didn't you? Most people do. And of course the two offices will continue to be held by the same person, for the same reason that Britain will continue to have a monarchy: no one can agree on anything better.
The only purpose of the Commonwealth is to hold the Commonwealth Games. There is nothing wrong with that, of course. But there is nothing more to it than that, either. You do not ever have to have been part of the British Empire in order to be in it. You do not in principle have to be in it in order to have the Queen as Head of State. Wars between its members, even with the Queen as Head of State on both sides, are far from unheard of. It has never been a single trading bloc.
As for "the Anglosphere", let's not even get started. Purportedly Tory Anglospherists should consider very carefully that by far the hardest place to sell the idea would be Canada. In India, the culturally Anglophile Indian elite of the immediate post-independence decades was never representative, and it is now a thing of the past.
In any case, consider well over a hundred years' worth of anti-British protectionist politicians on the monarchist Australian Right (legendarily the hardest people with whom British Ministers ever have to negotiate) at least as much as on the Left. Or consider the Canadians, who banned the acceptance of British peerages and honours in 1919, who enforced that against Conrad Black in 2001 and thus required him to relinquish his native Canadian citizenship, and who waited a full week after Britain had done so before declaring war on Germany in 1939. Next to those, Narendra Modi is a positively obsequious colonial satrap.
The New Right used to despise the Commonwealth. Powellites had principled objections, while Thatcherites regarded it as a nest of those who failed to show due deference to the Great She-Elephant. One shudders to think what Powell would have made of "the Anglosphere".