I have a flight to catch on 1 January at Gatwick airport. Can someone advise how I can go there from central London as I heard underground and easy bus will not be operating. Any other options?
I think the underground is running on 1st January. Tfl web page for 1/1/16 There are a couple of issues on Overground and Tfl Rail but they are outside zone 1 - the centre.
However this is not your problem. The problem is the mainline (National Rail) railway line to Gatwick is undergoing engineering work 24th Dec to 4th Jan 2016 so there are no direct trains from London to Gatwick. From Southern railway web site
Although Gatwick Express will not run during the festive period, travel by train to Gatwick Airport will be possible from London Victoria via Horsham, but this will take considerably longer (approximately 90 minutes instead of 30). Passengers are advised to leave more time for their journey.
Alternatively, passengers can travel by train to East Grinstead where replacement bus services will allow them to complete the journey to Gatwick Airport. (This entire journey will take approximately 90 minutes.)
So I would look at coaches e.g. EasyBus and National Express (as mentioned in other questions)and taxis/mini cabs
In additional to the two answers, National Express still operate regular coach services from Victoria coach station.
The trip takes ~1.5 hrs, cost 12 pound and schedule are as usual (~30mins).
On Monday 28 December, Tuesday 29 December, Wednesday 30 December, New Year's Eve and New Year's Day, the Tube will run a Saturday service with disruptions to some lines due to planned engineering works, so check before travelling. Don't forget, there's free travel on New Year's Eve from 11.45pm to 4.30am.
So you just need to visit TFL, plug in where you're going from and to Gatwick, and voila, you'll have the routes.
Failing that, you could bus, or Easybus (who claim weaselly that some trains aren't running, but that they will be!).