Wikipedia's claiming:
In addition, a number of other nations sent along small contingents of medical, transportation, construction and other experts as well as material help to the South Vietnamese government in
Saigon.
[8][9] These nations were:
Afghanistan,
Argentina,
Belgium,
Brazil,
Canada,
Costa Rica,
Denmark,
Ecuador,
France,
Greece,
Guatemala,
Honduras,
Iran,
Ireland,
Italy,
Japan,
Laos,
Liberia,
Luxembourg,
Malaysia,
Morocco,
Netherlands,
Norway,
Pakistan,
South Africa,
Spain,
Switzerland,
Taiwan,
Tunisia,
Turkey,
United Kingdom,
Uruguay,
Venezuela and
West Germany.
[8]
which is quite a few countries to pole at, but the source seems a bit obscure.
Here's an academic report on Many Flags and the Commonwealth that does confirm Canada and the UK sent some aid to South Vietnam, while also that Canada couldn't send troops due to being in the
International Control Commission.
It also makes it clear you are not getting the UK to join in if Wilson is PM. There is, however, an interesting bit that in a 1965 Commons debate, there was some support in Commons that compared Vietnam to the Malayan Emergency (possibly because of a recent bombing in Saigon). The writer is arguing our attempts to get into the EEC were an influence on Vietnam policy too, not wanting to seem too tied to America like De Gaulle muttered; the Foreign Office wanted to further back America and there was concern of losong that relationship; and one of Wilson's personal aims was to show he was a great international statesman and Britain was still a power player, and therefore he was going to show he was a proper independent head-of-state*.
So it could be more possible than we like to think that the UK would've joined up in a limited way under a different PM, possibly even a different Labour PM from the right wing but more likely the Conservatives if still in then, depending on which PM, their views on EEC, and how worried they are about the Special Relationship. And we'd surely present the war as Malaya II. Domestically, that could possibly be spinned after as "it
should've been like the War We Won but
the Yanks, eh?" and be a lingering black mark on US-UK relations (OOOPS!).
* Wilson thought he had influence, LBJ had "disdain for the man on a personal level" and the CIA thought he was dodgy. Though 'LBJ hated Wilson for not going along with things' does mean Wilson
was showing he was an independent head-of-state, so if you summoned his ghost he'd probably say "fuck Johnson, and also fuck that President from 1964-68"