My general understanding is that, weather-wise, where Americans would talk about it being “a nice day,” British people would refer to “a fine day.”
I still remember my first awareness of the latter. It came nearly fifty years ago, when I read Virginia Woolf’s To the Lighthouse. Looking at a Gutenberg edition of the novel now, I can see why I was struck by this usage. The very first line is, “‘Yes, of course, if it’s fine tomorrow,’ said Mrs. Ramsay.” The conversation continues (I have left out a lot of words about what Mrs. Ramsay’s son James, who really wants to go to the lighthouse, is thinking):
“But,” said his father, stopping in front of the drawing-room window, “it won’t be fine.”
“But it may be fine—I expect it will be fine,” said Mrs. Ramsay.
Now, Americans would sometimes refer to a “fine day” — indeed, a classic song by Carole King and Gerry Goffin says, “One fine day, you’ll look at me/And you will know our love was meant to be.” But the difference seems to be that American “fine” and “nice” bring with them a positive association, and are used in contexts other than weather. (“Have a nice day!”) Whereas British “fine” is more purely a description of weather we might call “fair.” Here’s an OED citation from 1913: J. G. Wood Insects at Home iii. 337 “On a fine day, it is very interesting to watch the ants.”
After all these years, I just encountered for the first time an American use of this “fine.” It occurs in Erik Larson’s The Splendid and the Vile: A Saga of Churchill, Family, and Defiance During the Blitz. On p. 44 we read, “Beaverbook convened key meetings in his downstairs library or, on fine days, outside on a balcony outside his first-floor bedroom (the second floor in American parlance.”
I have categorized this under “Ventriloquism” — the phenomenon of Americans consciously or unconsciously adopting Britishisms when writing about British people or topics.