Yeah, no. Many museums and galleries etc are closed on New Year's Day. The British Museum, National Gallery etc. You will find many shops won't open until midday and there are shortened hours for sights and shops on NYE.
I did fireworks on the Thames thing a few times, including for the Millennium, and it isn't a pleasant. A large number of bars will be charging a cover at the door, it is rammed, there are queues for the gross portable toilets.
Westminster Bridge is fine but is it like being in a cattle crush and you'll need to get there early for a good spot and don't think about trying to leave your spot and come back. The more pleasant experience was Blackfriars Bridge. Less crowded, good view along the river.