How would it happen? What about the Australian Task Force Vietnam becoming the Commonwealth Division, the RAAF bomber sqn becoming the Commonwealth Wing and the RAN ship becoming the Commonwealth Flotilla? Rember that both Australia and Britain were up to their necks in Borneo in 1965-6 when Australia comitted it's troops to Vietnam, Britain was running 5 brigades.
Involvement in Vietnam would probably come too late to save TSR2 and CVA01, but could see V bombers and Buccaneers in action.
I have seen threads on other forums about Britain sending Ark and/or Eagle to Vietnam. Possibly too late to save CVA01, but might cause a rethink of abandoning carrier aviation resulting in the Invincibles being Centaur-sized or slightly larger strike carriers.
Could the economy even support that, though, given that the Americans (and we ourselves) wouldn't be too hot on abandoning the emphasis on antisubmarine warfare vs the Soviets in the GIUK Gap either?
If the US forces drew on the experience of the British in Malaya and Borneo it could dramatically change the conduct of the war.
I think no matter who is PM, Britain is going to have to pretty much dragged kicking into it.