I'm...not so sure. The UK was a hugely proud nation in that era, and the general characteristic of the British when knocked militarily (possible exception of the post-WW1 world) is to respond by trying to work furiously at bouncing back from their loss. In addition, though this loss would be outside of the European "zone", British politics was almost entirely European-focused - the USA was seen as "just another country" not because they underestimated its strength but because it didn't get involved in European politics, which pretty much relegated it to being a second class state in the world at that time, and Europe really wasn't interested in involving the USA. Any notion of sacrificing independence to the USA would be seen by the UK and by Europe as a withdrawal from European politics, and this would be tantamount to political suicide. Losing to the USA in a war, even a devastating war, wouldn't finish the UK off, but doing this would pretty much be read as a sign that the British had given up trying. It would be like Germany in 1920 requesting annexation from France because they couldn't bear the shame of having lost a war. It would be a huge overreaction.
Also, I think there's a general perception, particularly from American posters here (not that that should be read into any more than because of your national history), that merging countries or requesting annexation was a popular or common idea in history. It really wasn't. While thinking outside the box is a praised skill in current times, in those days it was virtually unthinkable to do anything except follow current courtesies. The prevailing conservative mindsets lauded those who governed their business (particularly in government) by doing the usual things. Thinking up unique ideas was frowned up, and lead to ostracism. Even if the UK did want to collectively hang up its boots and give up on politics, the idea of offering to give up independence would just have never occurred to anyone, because it wasn't a "natural" thing to do.
It would be an interesting TL to write (maybe not to some of us proud British posters
) but this idea would be pretty ASB...