New England's likely gonna have a very similar issue as OTL's Turkey. There's precedent for the armed forces seeing themselves as the defenders of the nation's liberal, democratic values against authoritarianism, both foreign (the *US) and domestic (home-grown vitalists), including a Cincinnatus streak of having the right to overthrow leaders who they see as autocrats in order to restore democracy, not unlike how the OTL Turkish military sees itself as the defenders of secularism and Kemalism against Islamism. In New England politics, the "deep state" won't be a conspiracy theory, it will be a very real force with the military at its center. I expect the intelligence services and the bureaucracy to have a very similar attitude and align themselves with the military on the issue, especially once they're purged of vitalist apparatchiks.The United States, I would imagine, is liable to get even more authoritarian over time, not less; freedom of speech is still likely to be true but how much you get to say before you get reported to the authorities depends on where you live and this will be worse once the equivalent of international broadcasting comes about, to say nothing of the Internet. Death squads are something which have been used, for example, and enslaving even people that Americans ostensibly saw as their moral and political equals (i.e., white Canadians) is something that was eventually accepted as necessary to defend the security of the nation. Even if New England tries to thaw its very icy relationship with the U.S. while it rebuilds from Vitalist mismanagement, it has an ideological drive to try to use abolitionist efforts in its intelligence operations that it didn't necessarily possess before due to the state's socialist leanings. The Green Scare or whatever it's called is likely to be hundreds of times worse ITTL. Not only is New England a place with historical enmity working against them, but the fact that it speaks the same language means that any transnational literature will be scrutinized to the nth degree. It would be very easy to see how various departments can smear otherwise well-meaning Americans who want to see slavery done away with as Yankee, Australian, or any other sort of foreign-backed saboteur. The smart ones will buy one-way tickets while those that aren't so smart will find themselves sent off to some unnamed black site.
It would appear the way New England's government is set up is probably going to get a massive overhaul; we've gotten hints of this, like the governor-general seemingly being what amounts to a largely inoffensive figurehead. A multi-party parliamentary democracy in the style of Europe or one of the Restored Empire nations would probably be viewed as a safeguard from letting something like the Mullins regime happen again. There's also probably the perceived benefit of further casting away lingering American influence on their politics from New England's founding, though that's less practical and more for nationalistic reasons. That said, if there is a downside, it could go either way in regards to giving the armed forces permission to remove people perceived as too threatening to the Commonwealth. Now that a coup has happened, and its benefit was a net positive from what we can glean, that genie can't be put back into its bottle.
My personal head canon was always that the Vitalists ran away with their tails between their legs, assuming they didn't get caught, and left for Liberia ala the Nazis going to South America following WWII. I suspect they'd have golden parachutes when they got to Wilkinston anyway since Duvalier was so cozy with them beforehand.
That said, there's an interesting wrinkle I see here. In the modern era, the military and intelligence aren't usually known as bastions of progressivism. In New England, however, the army is the force that overthrew a reactionary tyranny. What's more, for various reasons, New England likely isn't engaged in much foreign adventurism, and what it is engaged in is mostly in support of forces opposing the *US, a far-right, white supremacist state. I think it would be safe to say that the New England armed forces and intelligence services would be very progressive and egalitarian on a variety of social and cultural issues, chiefly when it comes to race but not exclusively so, especially as time goes on. For instance, given the historic attachment of OTL's feminism and LGBT+ rights to liberal and left-wing political causes, I can see New England being one of the first nations to let women and openly gay people serve, including in combat roles.
Taken together, postwar New England is going to be a nation that is very, very culturally left-leaning on an institutional level, including in places that in OTL are bastions of conservatism in many nations, with social conservatives locked out of politics and regarded with suspicion as crypto-vitalists. In the postwar era, its culture will likely have a global reputation akin to a mix of West Berlin, hippie-era San Francisco, the Left Bank of Paris, and 2010s Tumblr, a place where the authorities are actively supporting ideas and activities that are considered countercultural and subversive in much of the world. There will be a backlash eventually, and given what I said earlier it's likely to get nasty, but the '40s and '50s in New York, Boston, Buffalo, and Detroit are probably gonna be wild, and Yankee movies and music will be treated the way OTL's Americans treated French and Swedish arthouse cinema, as either a much-desired forbidden fruit or sick filth that needs to be banned. It'll probably take a long time for the stereotype of the modest, no-fun-having Yankee Puritan to recover.
And in the other direction, to go back to what you said about rising authoritarianism in the *US, I can see the *US' previously libertine culture turning a lot more restrictive in response. They'll still be fine with drugs, for various reasons; if anything, there could be an "opiate of the masses" thing going there, as the government actively supports drugs not just as a black-market export but to keep the population docile. (That's why New Left activists criticized the hippie movement in OTL; Gil Scott-Heron, for instance, castigated them as drones who were no real threat to the system and would "plug in, turn on, and cop out" when the revolution came.) But like you said, there will be a very nasty alt-Red Scare seeing New England and Australia as subversive forces out to tear down American society. In a lot of more conservative quarters aligned with the government, cultural progressivism will likely be quickly identified with foreign enemies, as gender equality and gay rights are seen as slippery slopes to abolitionism -- and don't get them started on those who directly criticize how slaves and peons are treated.
OTOH, I can see the *US looking the other way towards a right-wing counterculture akin to OTL's alt-right. @BeyondTheBorg made an earlier post here about this, suggesting that, thanks to the *US' different historical experiences and the influence of Amber Jarrett, the mid-20th-century counterculture in their country could take the form of "an unholy philosophical amalgamation of Nietzsche, Ragnar Redbeard, Ayn Rand, the Nazis, BDSM, New Atheism, and alt-right chan culture", as well as ironically drawing influence from New England vitalism and suggesting that New England's backlash against it is a sign of their cultural and racial inferiority.