Should I postpone my trip to Paris in 2nd week of Jan in light of recent protests?

The UK government's advice to UK nationals visiting France is here: https://www.gov.uk/foreign-travel-advice/france

The relevant part says: "Protests against fuel prices continue across France, leading to blocked roads and motorways in some areas. You should avoid where possible and follow the advice of the authorities."

That is probably good advice for any visitor to any country experiencing some internal strife.

I used to live in France and never saw much in the way of protests except on TV. Protests in France (and elsewhere in Europe) can be noisy and are well covered on the news channels; and there are sometimes incidents of violence. But if you stay away from them, you can enjoy your trip in other parts of the city and barely know there was anything amiss.

The current situation in Paris, as of early December, is that there's a fairly large area on Paris that you should avoid on Saturdays. Everything is as usual on other days. (That's in Paris; in more rural areas, there are blocked roads. The blockades are mostly peaceful but they can make it harder to get around.) On Saturdays It's only a few hundreds of rioters amidst tens of thousands of peaceful protesters, but that's enough to be unpleasant if not dangerous if you get stuck between them and the police, and they sometimes move fast.

It may be difficult to understand which area to avoid as a non-local especially if you don't speak French; to get an idea, you can check which metro and bus lines are not running. That's in French, but all you really need to know is to avoid the neighborhood of the listed metro stations on the specified day. “Trafic interrompu entre … et …” means that the line is suspended between these stations. If a station in closed (“fermée”), you might see a mention “Les correspondances restent assurées” (the trains stop at an interchange station, but you can only change between lines, not enter or exit the station) or “Les correspondances ne sont pas assurées” (the trains will not stop).

That's the situation now. The situation is evolving rapidly, so it's impossible to make serious predictions as to what will happen in a month. However another month of protests at a scale that would affect tourists in urban areas would be extremely long, I don't find it very likely. Furthermore things are likely to quieten down for Christmas, and not likely to restart at the same scale. Road blockades may last longer but they wouldn't affect Paris.