Maybe Arthur is instituted as King instead of Governor-General (or is promoted in 1912/1913 time, just before WW1), and this practice is repeated for other dominions seeking independence.
I think you have four options
1880s - when the idea of an Imperial Britain was in full swing, drawing upon the German Empire, you could have the Empress Victoria of the British Empire and her descendants ruling sub-monarchies as King of Canada, Australia, New Zealand etc.
Post WW1 - I think maybe after WW1 is more likely than pre-war as the war is often viewed as being the point where the idea of being Canadian, Australian etc and detached (somewhat) from the UK became a much more widespread idea so the idea of having a separate monarch but one attached to the mother country may have been appealing. Of course in 1918/1919 at that time King George V had four healthy sons (Prince John, the fifth son died in 1919) and I anticipate he and Queen Mary may have felt more inclined to support their younger sons taking thrones as opposed to elderly uncles. Edward VII had much the same view on Norway, he pushed his son-in-law Carl of Denmark to accept the Norwegian throne as he preferred his descendants on the Norwegian throne as opposed to the other suggested candidate Carl of Sweden who was himself married to Edward's niece by marriage.
Post 1931 - As part of the Statute of Westminster some move towards independent monarchies under the headship of an Emperor (such title being held by the British monarch) as part of the gradual evolving nature of the Dominions relationship with the UK.
Post 1936 - the Commonwealth countries reacted more furiously to the idea of Edward VIII marrying Wallis Simpson than the British government. At that time, Henry of Gloucester and George of Kent were happily married to beautiful popular women and would have seemed potentially attractive candidates, its not inconceivable that infuriated by what was going on in London they wanted their own monarch and would look to the younger brothers of George VI. The Duke of Gloucester served as Governor General of Australia in the mid 1940s, Queen Mary's brother the Earl of Athlone acted as Governor General of Canada for the entire Second world war and had previously been GG of South Africa for over 5 years.