What did a New Year’s Eve Celebration look and feel like during the time of Jane Austen? This English Country House site gives you a good idea. You can still celebrate New Year’s Eve at Hunstrete House in Somerset (click on bold words) much as they did in the 18th & 19th Centuries.
One can imagine great logs burning in enormous fireplaces and a fabulous meal consisting of a variety of courses that stretched for hours on end. The evening would then culminate with the ringing in of the New Year and a festive group singing the Robert Burns version of Auld Lange Syne. (Listen to it on this site.)
In England, "If the family prefer to bring in the New Year at home there is such a custom: the members of the household sat themselves round the hearth, and when the hands of the clock approach the hour, the head of the family rises, goes to the front door, opens it wide, and holds it thus until the last stroke of midnight has died away. Having let the Old Year out and the New Year in, he shuts the door quietly and returns to the family circle. " (From this site)
The song, "Auld Lang Syne," is traditionally sung at the stroke of midnight in almost every English-speaking country in the world to bring in the new year. The custom of singing this song on New Years Eve goes back to the British Isles from the 18th century when guests ended a party standing in a circle and singing this song. The custom first was rooted in Scotland, because the lyrics were written in 1788 by Robert Burns, their favorite folk poet of the time. But most musicologists feel that Auld Lang Syne came from a traditional Scottish folk melody. The entire song's message merely means to just forget about the past and look ahead to the new year with hope. (From Study English Today)
More About Auld Lang Syne
The most commonly sung song for English-speakers on New Year's eve, "Auld Lang Syne" is an old Scottish song that was first published by the poet Robert Burns in the 1796 edition of the book, Scots Musical Museum. Burns transcribed it (and made some refinements to the lyrics) after he heard it sung by an old man from the Ayrshire area of Scotland, Burns's homeland," Borgna Brunner .