New Years: London or Paris with teens?

Just as alternative: Why not Berlin? It is one of Europe's largest NYE parties at Brandenburg Gate?
No real language barrier.
No Brexit risk.
A lot to do and experience for the kid and the young adult (in Germany >=18 years).
Both airports are well connected and good to reach on NYD.

Some tips and insights about NYE in Berlin.

As an American, flying in from the US, Brexit will have zero impact on you (if it goes ahead). If you don't intend on visiting anywhere outside of the UK then brush aside any concerns about Brexit.

Personally I would opt for London, much more of interest for all ages and a good NYE atmosphere.

Your kids bicker? Don't all siblings bicker? There has never been a trip abroad where my kids haven't bickered. I wouldn't drop the idea simply because of that.

Getting to the airport of New Years Day is possible, plenty of flights operate on that day. It'll be a bank holiday in the UK so public transport will be limited and taxi fares will be higher.

Mark from Berlin has suggested his hometown but as much as I like it I don't think it holds any comparison to London in terms of things to see and do.

@JC: imo Berlin can hold easily comparisons to London and Paris but is much cheaper. Anything specific you speak of?

Will brexit screw up a vacation in London by then?

How could that be?

Any preference for Paris or London at that time of year?

Can't go wrong with either.

If my kids bicker a lot is it worth trying to pull off this trip?

That is a surprising question. When we were growing up, if we bickered at a local restaurant, my parents left us home the next time the family was to go out for a meal. They didn't give in if we begged that we would be good. No way. They left us home. To go out for dinner was a PRIVILEGE and children who didn't appreciate that stayed home. If we bickered "a lot" as you described, my parents would never take out out to dinner let alone ON VACATION TO EUROPE. If your kids can't be well behaved since you described them as bickering "a lot", why even bother? You and your wife should go on your own together and tell them why they are staying home.

@JC: imo Berlin can hold easily comparisons to London and Paris but is much cheaper. Anything specific you speak of?

Berlin isn't much cheaper. I can easily eat, drink and sleep just as cheap in London than in Berlin but equally I can pay high prices in Berlin for average food. Berlin cannot hold a torch to London (or Paris) in terms of iconic sights, landmarks, museums, history. Sure it has its own attractions but not on the scale of London (and to a lesser degree, Paris).

Besides, the OP wanted advice regarding London OR Paris, he never requested any alternatives otherwise I'm sure the floodgates would have opened with suggestions. Personally I wouldn't want to spend NYE in London or Paris but that's my choice and I have my own personal reasons why I wouldn't want to, those reasons are, in general, not applicable to those who find the proposition quite attractive.

I would vote London. I love Paris (and have taken our kids), but I think London holds more interest for kids of those ages, between the shopping, museums, and shows. I also think London is a little easier to get around if you wanted to let older kids loose on their own. Have not been to Paris at Christmas time, but London is just beautiful that time of year with all the lights.

My kids bicker a lot at home, but I find when we're on an adventure together (like travel), the bickering goes way down. And would be easy to split up for a few hours or a day if someone is really getting on someone else's nerves.