If you want a pro British Dominion you need to look to New Zealand. Even post war the country prided itself on being Britain in the South Pacific. Jubilee year 1977 all we 7 year olds knew God Save the Queen, I don't think any of us knew God Defend New Zealand.
The trouble, much like with Australia, is that the Prime Ministers who happen to occupy office in all the Dominions - save South Africa - in the mid- to late-40s were, OTL, more interested in shaping the transition from Empire to Commonwealth. Perhaps that was a function of the realpolitik of the situation, but, given they were Liberal/Labor/Labour (Can, Aus, NZL respectively) I suspect it was actually more of a conviction than expediency. Interestingly, in each case they were replaced with PMs who radically disagreed with them.
Sidney Holland in New Zealand called himself a "Britisher through and through".
As noted previously, Robert Menzies in Australia described himself as "British to his bootstraps".
John Diefenbaker in Canada vehemently opposed the introduction of the Canadian Maple Leaf flag.
Which is why I'm asking this question regarding the Dominions, I suppose. In OTL, the Liberal/Labor/Labour PMs served concurrently with a Labour government in London who probably agreed with them on most things, including the reform of the Empire. How would they have instead got on with Churchill?