> It's not just the Greens - people who could be interested in joining a union but wouldn't want to fund Labour might include Socialists, Communists, Liberal Democrats, yes Conservatives as well (Conservative Workers & Trade Unionists for example), UKIP, and so on.

In the US union membership is a big issue. In the UK not so much. The unions do certainly care about membership, but they also care about actually having influence over the issues that matters to their members, as that's what they're for. At some point you need to choose if it's more important to represent everyone, or represent those who do join you well.

As it stands there's very little indication it'd be worth it for most UK unions to lose the influence they have in Labour. Consider that trade union support for Corbyn was greater than in Labours regular membership, for example - in other words, most of the unions have tended to lean to the left of Labour, not towards any of the other UK parties. Labour leaning UKIP members have largely returned to the fold, after the Brexit vote (and many places contributed to Labours result in the 2017 election). The far left have either embraced Labour again under Corbyn or at least accept it as an acceptable choice. Even TUSC and CPGB recommended people vote Labour most places.

This is a "problem" that mostly seem to be a problem to people who are usually not themselves unionised or interested in being unionised but who dislike the additional support it brings to Labour.

Put another way: If a substantial number of unionised people in the UK wanted unions affiliated with other parties, they can form them, or someone would form them and people would join. The Tories have tried at least twice (with CWTU that you mentioned being the latest incarnation) - it hasn't worked.

Instead what's happened after Corbyn is that unions that severed links with Labour during Blair has re-affiliated (most notably the Fire Brigades Union). RMT has not reaffiliated, but instead supports TUSC, to the left of Labour, but endorsed Corbyn in the Labour leadership elections. There are still other unaffiliated union options as well, as well as options to reserve yourself against affiliation in most unions affiliated to Labour (which means your money doesn't go to Labour and you don't get a vote in Labour leadership elections), so this is an artificial concern.

But if anything, UK unions have consistently been to the left of mainstream UK politics, even when some of their members end up voting for parties actively opposed to the same interests they've voted for when electing their union membership.

The proportion of members that reserve themselves against affiliation with Labour has typically also been low.

> Conservative Workers & Trade Unionists

... is a group set up by the Tory party as an attempt at trying to win over union support from Labour, that has totally failed at getting any traction, because it was started from the top with basically no bottom-up support. It was presented in 2015 as a major thing, and then quietly shoved under the carpet when they didn't manage to get anywhere.

One thing is conspicuously absent from their gallery page [1] for instance: More than a handful actual workers and regular members, as opposed to Tory party staffers, MPs, MSPs, MEPs and Lords. If they can't muster a room full of actual workers to front a "union", that say it all.

If they succeed at organising workers that feel unrepresented by Labour-affiliated unions in sufficient numbers to actually influence Tory policy, then great. But so far it has been a PR stunt from the Tory party.

To put it in other terms: The largest Tory led "union" was Conservative Trade Unionists (now "Conservatives at Work"), which at it's peak had at most a few tens of thousands of members, at a time when there were more than 10m unionised workers in the UK. It's been in steady decline since.

[1] http://www.toryworkers.co.uk/gallery/