I was thinking of a scenario earlier in which the American War of Independence either fails or is averted completely, could the United Kingdom restructure or reform to include British North America as an integral part of the country? I'm thinking of the "United Kingdom of Great Britain, Ireland, and North America" or just a more flavourful "Empire of the Britons" (Colonial Americans were white and spoke english, so close enough).
If you're talking about union along the lines of the 1707 or 1801 acts for Scotland and Ireland, than no. The colonies had long built up local structures for representative governance and would be unprepared to lose these for unification in a single parliament, as happened in the cases of Ireland and Scotland. Even if you set aside concerns about the colonists being overruled by the larger population in Britain, the logistics simply wouldn't work for day to day governance of the colonies. It took about three months to get a ship back and forth across the Atlantic, so it would be a very unresponsive government even on the most basic of matters that do not require deliberation.
What is potentially possible is some form of representation in an imperial parliament at Westminster, on top of the colonial assemblies. While the colonists would have accepted theoretical sovereignty lying with the imperial parliament, there would have to be de facto recognition by the British that colonial assemblies would handle most matters. In return, the assemblies would have to pay for the common defence, but would require more voice on these matters in the long term for this to be a lasting settlement.
All of this would be contingent on getting a different government at Westminster, and probably a different King, to the one we had in our timeline.