It could certainly be possible, but would probably require a radically different 20th century where the British Empire doesn't suffer the devastating economic, social and psychological effects of WWI.
Say the Entente Cordiale never happens; Britain isn't drawn into the conflict between the Central Powers and the Franco-Russian Entente. The Central Powers win, but the toll that victory exerts destroys Austria-Hungary and causes enormous ructions in the German Empire. In order to maintain its colossal European hegemony - a domination of a continent from La Coruna to the Urals, won at an unimaginable cost in blood and treasure - the Germans take a hard turn against democracy and shut the British and Americans (who both made fortunes selling weapons to both sides) out of their markets.
The British and Americans now find themselves ideological and economic allies against unFascist GrossEuropa (with an option on Communist governments in China and trans-Uralic Russia). The British look at converting their colonial empire into an EEC-like affair with added mutual defence, but when Mexico falls to a German-backed coup the Americans decide that 'united we stand' trumps '1776 forever' and petition to get involved.
Of course, the Americans still carry enough weight that calling it 'British' or 'Imperial' anything isn't going to fly, but a compromise is reached by calling it the Commonwealth of Nations; republican enough for Washington, British enough for the Imperialists and egalitarian enough for the socialists. Within a decade of its founding, it also includes those nations nervously clinging to the rim of GrossEuropa (Portugal, Greece, Norway) along with US client states in the Caribbean, Pacific and Latin America. There's even tentative ties with Italy, growing increasingly uncomfortable with German hegemony and encroachment on the Adriatic littoral.
And that's when the next war breaks out.
So, not quite the OP fulfilled outright, but close enough, I hope, for most people's liking.