Excepting the obvious POD...
With the exception of somesort of pre-American Revolutionary Imperial representation model, perhaps with Pitt the Elder in power, where the colonies are made self-governing provences of a Federal UK with seats in Westminster, there are few options. Of course, that is not really GB becoming a US state.
Ironically, the best possibility for a post-1789 Act of Union is a successful Nazi invasion of England after 1941. With the US in the war, perhaps Churchill, running a government in exile could come together with Roosevelt in some sort of transnational unity. It MIGHT be sold as a plan to unite not only GB but Canada as well. (Whether the Canadians would go along with such a notion is entirely up for speculation. Probably not. If there is one definition of Canadian, it is NOT American.)
Still, Roosevelt could theoretically offer Churchill the War Department or some other post as a sign of unity. Maybe even the Vice Presidency as his mother was an American Citizen, and Churchill could have qualified for the Presidency as surely as Obama has.
The constitution would have to be amended, but Americans have always had a fascination with Monarchy. They might accept a symbolic head of state.
Of course, under regular circumstances, no amendment of that sort would be entertained by Congress or the state legislatures. But, with the Nazi menace looming and notion of the English-Speaking Peoples standing together, it is possible that GB could be admitted as a state (under enemy occupation).
After the war, some kind of union might endure. Churchill dreamed of a dollar sterling, and his vision of currency union might have succeeded at the Brenton Woods conference were he still holding power after th '45 elections.
What would become of the colonies is another issue, of course. There would be little American support for making India or the African territories part of the Union. Still too much racial opposition. But, should they remain commonweaths as Puerto Rico has--at least until the late 1960's, maybe.
By that point, though, most of the colonies would prefer to become nations.