What if the Conservatives carried on running the country in the 1960s and we helped the Americans in the Vietnam war...
Okay, first off sentences are your friend.

That aside I think British deployments to Vietnam are certainly possible with the right conditions, see my answer to David Flin below.
... however we lose the war and it humiliates Britain on the world stage even further, worst than the Suez canal crisis in 1957...
This bit is harder as I can't see Britain sticking around until the very end, just like the Australians I'd expect to see troop drawdowns being announced in the late 1960s and starting at the beginning of the 1970s. During the deployment losing badly is tricky since they likely wouldn't be deployed anywhere too threatening to begin with, British troops having experience of the type of fighting, and no doubt demanding to run things their own way. Now warfare is a complicated affair so even with the best planning there might be a company or at most battalion that has bad luck and gets chewed up but nothing on the scale of Suez.
... the Tories are ousted in 1974 finally and Labour gets in (lead by Wilson or maybe Tony Benn) and takes Britain, a socialist Britain into the late 1970s and well into the 1980s...
Without Profumo there were still the other things, the government was starting to look a bit tired, and people in general start getting tired of parties that have been in power after ten-fifteen years or so. The easiest point of divergence I can think of is that the Conservatives are able to win the 1964 general election, it was pretty close in a number of places, with Home sending a token force to South Vietnam.
Problem then becomes 1969, there's simply no way that the Conservatives will win that general election after eighteen years in power. This in turn means that Labour will now be pretty much guaranteed to be in power for at least a decade, which considering this is the 1970s we're talking about is practically the dictionary definition of a poisoned chalice. I honestly can't see how you can stretch out a Conservative government past the 1969 general election short of adding large doses of lead to the tea supply at Transport House and making Labour do something unimaginably stupid.
Douglas-Home was influenced by Powell, who wouldn't have pissed on the Americans if they were on fire, so there is not a hope in hell of Home taking Britain into Vietnam.
Home
was very anti-communist. If the Americans were to open the chequebook to help fund it, and support the wider British economy, I could perhaps see them sending an infantry brigade and support to run a small quiet province as a gesture. Dress it up as a request from the South Vietnamese and from the Australians to play up the Commonwealth angle, just as the Australian deployment was nominally under American command but in reality had a direct line the Westmoreland doing what they wanted have the British contingent come under the Australians but really be independent. The airfield complicates things but Bien Hoa province, neighbouring the Australians, looks like a half-decent option if you could swing it. Of course everyone would know that it was really being done in support of the US but it would give them at least a thin cover, the people that protested the Americans having a domestic target for their ire in the UK would also likely be more vigorous.
Assuming that American financial support is able to paper over the cracks that could potentially mean a less serious balance of payments crisis allowing devaluation to be put off until after the general election. Speaking of that I'd expect the Conservatives to announce the drawdown of troops in early 1969 well advance of it, helps defuse the issue slightly and quite frankly they'd need all the help they could get.
Even though the premise here is flawed, I suppose it's worth addressing the fact that the Tories winning in 1964 would still have been a close-run thing and that Wilson would have been PM by '68 or '69.
A change of roughly 5,800 votes in the right twenty constituencies would have given the Conservatives a majority of 17, change roughly 3,660 votes in 14 constituencies would still give them a majority of 5, which without looking to see how many MPs died between 1964 and 1969 might just be enough to scrape them over the line.