One idea I had a while back was for the Quebec Boundaries Extension Act 1912 to not happen with the land from the North-West Territories east of the Hudson Bay going to Newfoundland and Labrador, no idea how to get it to happen with a point of divergence after 1900 though. Wouldn't have much of an affect to begin with but long-term that region is where most of the large hydro-electric projects were built which if they were able to raise the capital would provide a decent income for them.
That would create a dispute with Canada, for one. That's because that area is part of the NWT as the District of Ungava - an area of Rupert's Land that has been Canadian since 1870, when it got purchased from the Hudson's Bay Company. If Britain/Newfoundland wanted to go for that, they could, but Canada would want something in return, even if it is financial compensation. You'd still have the boundary dispute with Canada (= Québec) over Labrador in general, with Québec insisting that Newfoundland holds no claim while Canada would hold to the more "moderate" position of Newfoundland being restricted to the coastal areas only, so extending it that far would have all sorts of repercussions.
Finally, you'd still have a large Aboriginal population, most who would still speak their original language at a far higher rate than elsewhere in the UK - while most of them would be largely Inuttitut-speaking (the local pronunciation for
Inuktitut) and in that case I could see synchronic digraphia like Bosnian/Croatian/Serbian/Montenegrin which most focus going towards
Syllabics alongside the German-derived morphophonemic Latin orthography (based on that historically used for Greenlandic until the 1970s) introduced by the Moravian Church, there are also some communities which speak either Cree or Innu-aimun/Naskapi - the former along the Hudson Bay coast (as part of their homeland, which they call Eeyou Istchee) from Eastmain to Whapmagoostui (Waskaganish would remain Canadian) and the latter in the interior. Historically, language policy in Newfoundland was not very kind to minority languages, as least on the Rock proper, but in Labrador (at least) the indigenous languages were somewhat better preserved (but still in a precarious state). So both Native policy and language policy would be something that the UK would have to improve on, considering that Britain directly never got itself involved with it except with processing the treaties that were signed in the Crown's name.
Still, though, it would be something to see - the road not taken. Would make for something interesting, considering the whole place is at least 3x the size of England within OTL boundaries, so would go for an interesting counter-weight to Ireland and Scotland - particularly the former since much of the Newfoundland population originated from southeastern Ireland (primarily the area around Waterford) and southwestern England (primarily Devon, Dorset, and Hampshire).
In the event that it was a possibility, would the US kick up a fuss about integrated British territory in the Americas? Or were relations with the UK friendly enough by the 20th century for it not to be an issue?
That would be interesting to see. In reality, I would assume that Washington would not put up a fuss - too far away from the rest of the States and too significantly unimportant for anyone to be bothered. Not based on friendliness with the UK, but on pragmatics and practicality. Now, if we're talking the mid-20th century, the only thing the US would be worried about would be continued access to those bases set up during WW2 (i.e. Gander, Argentina, Stephenville, Goose Bay, etc.).