While from the forum point of view this is where the Thames output finished and Carlton started, its unique from the other point of view that the entire concept of celebrating New Year was fairly low key - a large crowd gathered outside Big Ben to see the chimes and that was pretty much it - and Chris Tarrant having the honour of being the first in-vision presenter on the new station.
In those days there wasn't any special in London to mark the New Year. I remember the year after 2000 Today, the Beeb did a live lottery show across midnight with reporters across the UK, and in London there was Bradley Walsh with about half a dozen people, with nothing partcularly happening.
Yes, there was no official celebration in London. The tradition was to congregate in Trafaglar Square but even that was warned against by the authorities and there were no extra facilities put on for it. They were just standing in the square, the fountains and the lions were boarded up and it wasn't made particularly welcoming
2000 changed all that - the focus moved to the river and fireworks and it became the proper organised, ticketed event it is today. I think 2000 was the first year that TfL made the tubes and buses free on NYE as well
The New Year's Eve fireworks in London didn't really get going until 2003/4 - the 2000/1 event was cancelled as Ken Livingstone couldn't persuade London Underground, which the government wouldn't let him control because it wanted to implement a part-privatisation deal on it, to run an all-night service. http://news.bbc.co.uk/1/hi/uk/1032426.stm
The same applied for the next two years, until Livingstone got control of the Tube in 2003, sacked the senior management, and fireworks and an all-night service began. The fireworks display became ticketed under Johnson in 2014: https://www.bbc.co.uk/news/uk-england-london-29234879
You can see the Trafalgar Square "celebration" of 1992/3 (together with a scrolling "Happy New Year from the Metropolitan Police" message) in the opening minutes of A Carlton New Year: https://www.youtube.com/watch?v=3zRnRUOLDgk
And this was what this year in the square was like... https://diamondgeezer.blogspot.com/2020/01/2020-in-trafalgar-square.html