I think it's more a matter of terminology - calling it the EU carries all sorts of implications from Euro-centricity to the way it is governed to being an economic alliance with no military content.
That's fair enough. One thing to note on this, as it's a mistake I've seen made before, is that the EU is a military alliance. The
WEU existed from 1954-2011, when it was subsumed into the EU. Part of being in the EU is signing up to the mutual defence clause, which obliges members to use all means (implicitly including WMD), to protect other members from attack.
All of those aren't sustainable ITTL - the remnants of Empire just have too strong a pull, and whatever it is will be growing out of a wartime military alliance rather than postwar coal allocations.
I agree it will be different, but, in some terms it will also already be deeper because of the wartime experience. The Franco-British plan was for complete economic integration during wartime, to essentially operate as a single economy. This will, I think, be extended to the part-occupied minor allies simply be necessity.
When Germany is defeated, it will then be very hard to break apart the now integrated economy - and there will be a very good reason not to. German's erstwhile ally and later enemy, the USSR, will be sitting there to the east, occupying Poland and who knows what else.
Unlike iOTL, where it had been an ally for several years, and there was a lot of wishful thinking that it would remain so, here it's never been more than a co-belligerent opportunist that stabbed its own ally in the back for a share of the spoils of a victory it did nothing to earn. It's also going to be a lot more obviously powerful.
As a result, I think the wartime economic integration will be continued, simply to allow them to stand up to the colossus that bestrides Eastern Europe.
As a result, you'd be starting from a position of free movement of goods, capital, services, and labour within the Western Allies. That's the baseline, which I think will continue.
The question of free movement of colonial labour (no one will care about it being extended to the white dominions) will be more challenging, but I think in no way insurmountable. Just about all the participants already have colonies, so already regulate (or don't) the movement of non-Europeans they considered undesirable.
This is before the era of mass migration, so it won't be recognised for the political problem it will later become, as it isn't a concern for another decade. Later on, when it does, free movement of labour simply wouldn't be extended beyond Europeans. Racist, but not exactly hard.
There is another potential factor here. As economic integration will continue, then post-war labour shortages in the UK will probably be met by importing European workers, rather than commonwealth ones. They're just that much closer.
It's fairly clear that the British and French are already thinking in terms of a very much stronger postwar alliance between their two countries - the French in particular have had a bad scare and are well aware that they only survived thanks to the British, while the British are looking at the size of the Wehrmacht and know they couldn't hope to beat it by themselves.
Less the size of the Wehrmact, and more the size of the Red Army, I think.
Whoever else joins (Norway is almost certain to, while the Benelux countries I'm less sure about - they may follow the Benelux route and opt for some sort of mutual defence pact with the Entente), they're going to have to fit in with an agenda set by London and Paris because they're simply too weak to set it themselves, even when clubbing together as a bloc (one of the potential drivers for Benelux ITTL, incidentally).
After what Belgium got up to inter-war, I don't see the French accepting this. They're going to want something much more binding that they can't back out of so easily.
They will also be in need of Entente support to help rebuild their economies (and feed their population) after the devastation of war and occupation.
Who else joins depends very much on how the war ends, and where the Soviets end up.
Well, one for you to know and us to find out, I suppose.