The challenge is to make all of the former British colonies retain their status as Dominions instead of turning into republics. In 21st century they're fully independent, though, but shared with the United Kingdom the same person as their respective monarch.
The most obvious probabilities are India, Pakistan, Ceylon (Sri Lanka), Kenya, and Nigeria, since IIRC all of them were granted Dominion status following the WWII, but soon became republics.
And what would be the consequences? Larger Commonwealth?
PS: I post this in pre-1900 forum because I think this will require pre-1900 PODs.