Can I go from Gatwick Airport station to London Bridge station?
I have read that the train will not stop by London bridge due to construction, is this true?
I am travelling by the end of January.
The easiest way to answer this is to check the Transport for London Journey Planner, this will show the various ways you can get there and also highlight any (planned) closures, construction, etc.
Running Gatwick to London Bridge through that it suggests:
17 mins (national-rail) Southern to East Croydon Rail Station
19 mins (national-rail) Southern to London Bridge Rail Station
46 mins (national-rail) Thameslink to Blackfriars Rail Station
4 mins (tube) District line or Circle line to Monument Underground Station
Disruption: Part Closure
4 mins (walking) Walk to Bank
1 min (tube) Northern line to London Bridge Underground Station
There's also the Gatwick Express which goes from Gatwick to Victoria (faster but more expensive). There are various ways to go from Victoria to London Bridge. I think there is also a direct Southern train between Gatwick and London Bridge but it's not showing up in the search.
But just to double check the construction, you can check the London Bridge travel advice page, the key point:
How your travel could be affected
London Bridge National Rail station will remain open and London Bridge Tube station will operate as normal throughout the rebuild. However London Overground, Southern, Thameslink and Southeastern passengers will be affected by the improvement plans at certain times up to 2018. As a result, some services and stations on the Tube, London Overground and National Rail networks will be busier than usual as National Rail customers use alternative routes to complete their journeys.
The next period of major changes to services is from January 2015 continuing to August 2016.
From Monday 5 January 2015, Southern services will start to operate to and from London Bridge again, with a new timetable across the Southern network. Thameslink trains will run a limited service to and from London Bridge until January 2018. From 12 January 2015 to August 2016, all Southeastern services to and from Charing Cross will not call at London Bridge.
That shouldn't impact the journeys above (apart from the limited service on Thameslink). However, the Southern site also says:
London Bridge: Southern and Network Rail apologise for the disruption and delays at London Bridge recently. The new track layout on the approach to the station has meant that so far, it has been a challenge to deliver the new timetable. We have reviewed this and made some changes to a small number of evening peak services to help us to deliver an improved service for you. We will be monitoring the service at London Bridge and assess if any long term changes to train services are required.
So it's definitely going to be worth double checking things closer to the (via the links above or listen to the announcements, read the info boards and ask the station staff).
There are also buses from Gatwick, but none get you much closer than the train.
As SpaceDog mentions, there is currently a lot of disruption at London Bridge station at the moment.
I'd recommend getting the Gatwick Express straight to Victoria (or a normal Southern Trains service if you don't want to pay the extra for the express), then getting the District or Circle line to Monument, and walking over the bridge. There's no point walking from Monument to Bank and getting the Northern line as TFL suggest, as you'd spend more time going up and down escalators than you would walking over the bridge!