Crudely: why on Earth would they?

Less crudely: it depends what you mean by support

I can speak for the UK side of the question here. Britain in the 1960s was a fulcrum of pressures, failures and new politics, each bouncing off one another. International decline, uneasy sense of subordination to the US foreign policy, economic stagnation, generational dissonance, ideological staleness – 1960s Britain had it all.

The captain of this unhappy ship in the period where Vietnam policy became crystallised was PM Harold Wilson. Put simply the UK’s calculus on this issue is pretty watertight, and a bad hand is played well (or at least better than other areas of Wilson’s premiership).

Essentially Wilson had little time to play the international statesman. A boy of the traditional left of Labour, who moved to the centre discreetly enough through the late 50s to keep his personal ideology obscured (until decisions had to be made that is) and clinging on to a slender 5 (shortly 4) seat majority. His pitch: ‘white heat’ technology-will-save-us-from-structural-decline industry 2.0 esque, (which the cynic in me puts as a very well-crafted political messaging campaign with little detailed planning, and even less actualisation given Wilson’s paranoia and clipped cabinet). He faced a stagnating economy, an overvalued pound teetering in its position as the reserve currency and a nation morbidly facing its own decline, with a kaleidoscope of reactions by section, age and political persuasion. These problems are not helped by an over-costly empire, outsized defence spending and a systemic failure to tackle deep-seated problems in the UK economy.

Denis Healy, Minister of Defence, set a plan in motion to reduce the military budget to £2000m by 1970. £2000m. In a country which is in the middle of a decades long financial ruination. It was an expense Britain could barely afford, and which showed in military preparedness across the services. Simultaneously decolonisation was in full swing, with the 60s seeing the gradual transition from denial, anger, bargaining, depression and acceptance as the colonies slipped away/were granted freedom/freed themselves – however this process kept a weary alertness and readiness to deploy force in the adhoc basis the colonial policy had been running since 1945. Quite simply therefore, Britain could not afford any more exacting engagements internationally, indeed it couldn't afford its current buffet of self-created issues alone. To open an additional front, of questionable value (and highly questioned value in the Foreign Office) would be the height of folly. Particularly for a government which is about technocratic governance lifting the country out of economic malaise through 'new' politics and economics. To join America in Vietnam would be.... Churchillian in all the wrong/politically unhelpful ways.

Britain’s relationship with international Communism was also different, particularly under ‘socialist’ Wilson. While the Cold War fears were always there it always had a more defensive posture (perhaps bracketed by the economic weakness of the period and leadership (thus deferment to) US action. There was not ever really the same commitment to domino theory (beyond an Imperial-focused pale imitation a decade before). Philosophically Britain was about containment, at low expense using means which avoided violence it could ill-afford politically or economically. Vietnam, never being a UK colony did not even qualify for that vague reason to be actively interested in it. Thus there was not the same sense of mission (and subsequent gambler’s fallacy) to get involved in that particular distant country.

In addition, with the 1960s came the rise of a new generation of left-wingers, ones more concerned with international and social issues than those of the traditional left-winger Wilson. While much of their agitation happens through satellite groups, their loud discourse became increasingly conflated with the left as a whole, and began to reflect in membership and voting intentions in Britain. The membership of CND in particular, drifting since the Test Ban Treaty, came out in force against the Vietnam war, providing fresh blood and a broader base than the Trotskyites/Tariq Alis of the world and pressure upon the man of the young and man of the left Wilson. This issue considerably ate at his political image and legitimacy, and while doing little electoral damage directly, the PR side tarnished him early. This was without even going into conflict, without soldiers dying, without images on TV screens. To start a war would ratchet this distracting pressure up several dials - for 0 political gain. Wilson was a canny operator, very much a forerunner of the modern politician, he was too wise to take this gut blow for something as frivolous and empty as a duty to the 'special relationship'.

Now he has a problem, the US’ fiscal muscles by the mid 60’s are as ominous as they are obvious. Desperate to avoid devaluation (savings/pride/fear of inflation – the vices of Little England) support in some form from the US was needed. In private meetings (Dec ‘64-Sept ‘65) the Johnson administration pressured Wilson to send at least token forces to Vietnam. Wilson, staring down the at the meagre political hand drew a firm line at involvement in Vietnam for all the reasons mentioned. Politically, he faced hostility on his left, studied disinterest in the centre, and only muffled cries from the right (who preferred to badger him on British adventures, not yank ones). Militarily his forces were stretched, weak, and of questionable assistance in a conflict that experts in the UK quickly believed the US had no chance of winning (though the mandarins often drew very different arcs to get to that conclusion). Internationally Britain wasn’t the cold war warrior the US wanted it to be. And economically Britain could not afford the conflict. The trade was Johnson accepting this political reality (ultimately something he respected to a point – coupled with the fact that the US ‘didn’t give a damn about Wilson’ and the questionable value of the UK militarily) in return for Britain keeping its ruinous presence in the Suez and publically supporting the Vietnam war (if only rhetorically).

Seeing a good deal when it was presented Wilson agreed to this, to his increasing cost as the left piled into his ‘collaboration’. He kept this line broadly throughout his leadership, touching base with the left occasionally, for example his 1966 visit to the USSR PM Kosygin to ‘try and work a solution’. His reasoning varied, but never felt quite so honest as his famous retort to student protestors ‘we can't kick our creditors in the balls’.

TLDR: There was very little compelling reason for the UK to get involved and plenty of limits, with Wilson walking a tight line to provide rhetorical support (at great cost) and nothing else.