As the title says : How can have a Commonwealth still existing, but with sovereign of Canada, Australia, or other Commonwealth countries (having or not a monarchic ruler OTL) being distinct from the King/Queen of UK?
Well if the Aussie Referendum had gone the other way then Australia would have a different head of state than the Queen but would still be a member of the Commonwealth.
In resume : how having Canada or Australia having their monarch.
Perhaps divide up the Dominions among various brothers and sisters, rather than all being inherited by one heir. A generation or two down the line, you have several distinct royal families which all belong to the same house.
For example (referring to people by their OTL titles in bold):
- Queen Elizabeth II is the Queen of the UK plus various minor territories.
- Charles, Prince of Wales is the King of Canada -- a position that is held by the UK's heir apparent. His mother Elizabeth was previously Queen of Canada, and also Queen of Newfoundland until it joined Canada.
- Prince Richard, Duke of Gloucester is the King of Australia and the King of New Zealand -- positions he inherited from his father Henry.
- Prince Edward, Duke of Kent is the King of South Africa -- a position he inherited from his father George.
I can't see that as being acceptable to Canada. We already get too much of "you're ruled by the Queen of England?" type comments as is, never mind if the Canadian monarch was just seen as a waypoint to the UK crown. Repatriation of the monarchy has been mooted about in some circles as an answer to republican sentiment and I can see a Canadian monarch but not a "minor league" situation as you describe.