I had asked a British theater group before going to the show if there would be anything that I as an American would need to know, and they just vaguely mentioned things about the coal miners' strike and generational tensions, but said there wasn't anything major. Turns out, I felt pretty lost in the first half and was desperately googling at interval to try to keep up (I didn't realize that one of the elections mentioned was for Thatcher, or which war it was, or even that the Park Hill apartments were a notable apartment complex--also didn't know what "being listed" meant). There were also just moments where it was unclear to me what exactly had happened, and I wasn't sure if that was because I had some slight trouble keeping up with the accents, whether there was some specific reference I missed, or if it was just the writing itself that wasn't clear. Granted, you can still get a perfectly lovely story without knowing the full context, but I think if it were to come to Broadway it would need substantial rewrites to not leave the audience feeling like they're missing out on an inside joke (which I felt many times).
There's also the matter of "will it sell?" It seems like the album and artist that the songs come from are much more well-known in the UK (I hadn't heard of either of them at all before) so I don't think that would be a huge selling point in the way that Bob Dylan was for Girl from the North Country, and I can't imagine "a multigenerational story about an apartment complex in Sheffield, England" would necessarily generate as much interest either. And while other shows have had hyper-specific in-jokes before, it was also with the assumption that there would be a certain percentage of the audience that was guaranteed to get it (like some of the Broadway jokes in The Prom), and I'm not sure there would really be much or any of the audience on Broadway that would fully grasp the Sheffield-specific references on a nightly basis.
That said, I don't actually think things like the set would be a huge issue. Yes the set as it was on the West End is quite large, but I think there could easily be a scaled down version of the set as well. Most of the utilized space in the set really came down to an upper level and an outline on the floor, which could be done in almost any theater (though a bigger one would be preferable just to give a better sense of scale).