According to William Hone’s period tome, The Every Day Book, New Year’s Day in London was primarily celebrated with small social gatherings and the wearing of new clothes (out of a superstition that not doing so was unlucky—which Mr. Hone disparages). He also says, “The only open demonstration of joy in the metropolis, is the ringing of merry peals from the belfries of the numerous steeples, late on the eve of the new year, and until after the chimes of the clock have sounded its last hour.” I think I’d prefer that to fireworks.
The whole “Twelve Days of Christmas” song comes from the twelve days between Christmas Day and January 6th (Epiphany). In the Regency, Christmas was more of a religious celebration but Twelfth Night (either January 5th or January 6th—no one seems to agree which “night” it is) was a party. They had Twelfth Night Cake or what we call “king cake” in New Orleans. There were parlor games and balls, and a good time was had by all. Maybe that’s why the last four days of the song are about lords leaping, ladies dancing, pipers piping, and drummers drumming. Partay!!
For Twelfth Night in Regency England, the custom of choosing a king and queen from whomever got the bean and pea in the twelfth-cake evolved into choosing characters out of a hat to pretend to be for the evening, a sort of masquerade. In some cases, they wore masks and the person was required to remain in character the entire night. Sounds like fun to me.