The first impact will be the loss of services to the centre of town (Charing Cross) in January 2015 – either directly or by changing at London Bridge (because Charing Cross trains will not stop at London Bridge). The second major change will be in August 2016 with the loss of any access to London Bridge (and thus to all other places, such as Gatwick) until 'early' 2018.
12 January 2015 onwards (forever)
• Southeastern services from New Cross, St. Johns, Deptford, Greenwich, Maze Hill and Westcombe Park stations will no longer operate to and from Charing Cross or Waterloo East, and will instead operate to and from Cannon Street. These diverted services will still call at London Bridge
12 January 2015 to August 2016
• Southeastern services to and from Charing Cross will not call at London Bridge
• Southeastern services to and from Cannon Street will continue to call at London Bridge
August 2016 to early 2018
• Southeastern services to and from Charing Cross resume calling at London Bridge BUT
• Southeastern services to and from Cannon Street will not call at London Bridge. Some diversions will take place between Charing Cross and Cannon Street
• Southeastern services to and from Cannon Street will resume calling at London Bridge
For a full list of services affected by Network Rail's rebuilding of London Bridge (as a result of the Thameslink programme) go to this Transport for London page.
Update 28 Oct:
There is a campaign to reduce the disruption – see GLUG (Greenwich Line Users Group). Contact them via firstname.lastname@example.org.
Thanks to From the Murky Depths for further clarification on the decimation of our services:
A big issue is 4 trains cut in the evening peak (4-7pm) to Deptford, Greenwich, Maze Hill and Westcombe Park. Three are at the busiest time of 5:30 - 6:30pm. This is around 38 carriages in total, and 30 alone from 5:30-6:30pm.
They say they will extend other services to make it up but the numbers don't seem to add up to maintain capacity at the busiest times. Other trains are already around 8 carriages long, with some 10, and trains cannot be longer than 10 as Woolwich Dockyard along the line has short platforms, and no mitigating action has been taken about it.