Not much, to be honest. The fact is that the US already has good relationships with Spain/Cuba and Mexico in the Caribbean and doesn't really need another strategic partner in the region. The fact is that Puerto Rico's notice to withdraw is basically a threat that has got out of hand and, while there are certain members of the political elite in that country who do want to leave (it's not as if they've been treated well by that organisation in the past decade and as a Hispanophone country many people do feel culturally quite distinct from the rest of the Commonwealth) I think most of the political leaders are hoping that something comes up which means they can back out the whole process. Remember that the actual referendum wasn't explicitly one about leaving the Commonwealth as such.
Gee, why does that sound familiar...
Speaking of, something that I've been wondering about, and which this latest update put front and center, is whether we might see an analogue to OTL's Euroskepticism in the form of anti-Commonwealth sentiment. We're already seeing it in full flower in Puerto Rico, and I'm willing to bet that Newfoundland is also having second thoughts. What I'm thinking about is the Big Four, particularly the UK, especially if Thatcher's overreach leads to the UK's power over the Commonwealth being reduced in the future. I'm not expecting it to be too pronounced in Australia; there, I'm willing to bet that ore mining (uranium, iron, nickel, bauxite, copper) and agriculture are supplying the working class with all the well-paying, hard-to-outsource jobs they could ever want, and so anti-Commonwealth views are a fringe minority. Even in Canada, where there are undoubtedly more politicians who think that the Commonwealth gets in the way of a closer relationship with the US, you'll also have a massive natural resource industry (ores, grain, timber) to keep the working class from joining them -- and likely just as many politicians who are afraid of Canada becoming a vassal of their much larger neighbor to the south, and see the Commonwealth as an effective antidote. Pakistan, meanwhile, is among the clear winners; everybody there knows where their bread is buttered.
In the UK, however? Their only major natural resources are fish (the Commonwealth is likely gonna tell Iceland where to stuff it in the Cod War) and North Sea oil. Coal is on its way out thanks to the UK's embrace of nuclear power (fueled by Canadian and Australian uranium); they've been trying to build plants in former coal-mining regions in order to limit the damage to working-class communities, but I doubt that the skills to operate mining machinery translate all that readily to operating a nuclear power plant, and I think you've mentioned that there's still been a lot of dislocation. Manufacturing, meanwhile, is being outsourced to Pakistan, the workshop of the Commonwealth, and now that Pakistani wages and living standards are reaching British levels, I expect it to start going to Commonwealth Africa after that. Sure, the UK
as a whole is prospering, but there are some substantial sectors of the country that are being left behind, their livelihoods rendered obsolete to the point where, like you mentioned, some have emigrated to Pakistan looking for work. (On the same token, I imagine a lot of British coal miners winding up in Edmonton and Adelaide.) While you didn't bring it up in the 1976 election update, I think this may have been an unstated factor in the Liberals' victory: not only did the new middle class produced by the "new economy" have less affinity with working-class politics, but the traditional British working class, even after thirty years of Labour government, was seeing its position decline relative to the educated middle class, and their disillusionment with Labour caused them to either stay home or consider voting Liberal. (After all, the Whigs ain't the Tories, and they still have a tradition of standing up for the working man, right?)
I doubt that we'll ever see a full Brexit where the UK straight-up secedes from its own Commonwealth. As my phrasing there ("their own" Commonwealth) suggests, there is a ton more romantic attachment to the Commonwealth than there's ever been to the European Union; after all, there's direct continuity between the Commonwealth and the British Empire, the period of the UK's greatest glory, whereas the EU was a construct led by the French that initially had an almost purely economic purpose, the UK only joining later. And TTL's Britain still has a ton of real weight behind it as the core of one of the world's three superpowers, as opposed to its OTL post-Imperial decline to middle power status. Commonwealth skeptics in the UK will be like Euroskeptics in OTL Germany, in that the UK has too much to obviously lose from leaving and everything to gain by staying. However, I expect that, even if it's firmly rejected by the mainstream and never going to happen, the idea of pulling out of the Commonwealth will hold some appeal on the fringes of British political discourse. The far left, the ones who think Labour under Attlee sold out and that he should've nationalized the heavy industries, would accuse the Commonwealth of being the new Empire, an instrument for capitalist power with only a coating of democratic socialism to lubricate its sodomy of the working class whose jobs it now ships to Pakistani sweatshops (colorful language that will no doubt be used in the pages of TTL's version of the
Morning Star or
Living Marxism). Some of them might go as far as to point to Bengal and Puerto Rico as examples to follow; this would undoubtedly be a very sore point in the '70s, one that Labour will likely use to discredit anybody to their left, but if Bengal and Puerto Rico don't turn out to be total shitshows (even if they're just middling post-colonial states), then later on they might whitewash them and use them to bolster their case. The far right, meanwhile, would blame the Commonwealth for all the brown people in Britain and bemoan how London is no longer running the show and putting the "colonies" in their place like when they called it the Empire (especially if Thatcher's attempt to exert British control over the central bank backfires), and claim that migration within the Commonwealth is being used to destroy national identities that would offer resistance to the elite's rule. You already brought up that, in the 1962 Commonwealth elections, there was a far-right grouping in the Commonwealth Assembly led by A. K. Chesterton that managed just over five percent of the vote, campaigning on opposition to South Africa's expulsion. I'd imagine that, in later years as the non-white Commonwealth members grow increasingly powerful, the "rising tide of color" would become a talking point in the right-wing tabloid rags.
On a similar note, how's Quebec, particularly as far as separatism is concerned? The fact that we've heard nothing about it indicates that, like anti-Commonwealth sentiment, it's probably a fringe position in TTL's Quebec, especially as compared to OTL where there were two referendums on the matter (one of them a nail-biter). There are certainly a handful of people who dream of an independent Quebec, but they can't command any popular support. I imagine that the success of the Commonwealth has suppressed nationalist sentiment in Quebec -- in OTL, an independent Quebec would've merely gone from being part of America's hat to being America's
other hat, with little practical change in their status next to their most important ally, while ITTL, becoming a small nation of only six million people next to a superpower of over 200 million (using '70s census figures) would've meant a serious loss of status and prestige versus when they were part of one the Commonwealth's Big Four. And that's without PRexit going pear-shaped; if it does, then that would likely kill support for an independent Quebec stone-dead outside of a small fringe.