The street cabinet in Bermondsey Wall West has recently been upgraded to support high speed fibre broadband (e.g. BT Infinity), which should now be available to all residents of Jacob's Island.
We have a "fibre-to-the-cabinet" (FTTC) service - this means that the broadband connection uses very high speed fibre optic links from the BT exchange to the green cabinet in the street. The existing phone lines only have to carry the broadband signal from your flat as far as the green cabinet - since this is much closer than the Bermondsey telephone exchange (where the broadband signal terminates for existing ADSL/broadband services), much higher data rates are available.
A full list of service providers offering fibre broadband is at http://www.superfast-openreach.co.uk/buy-it-now/
A new Router/BT Home Hub is needed, to support the high speed (VDSL) link to the street cabinet; this should be provided by your broadband supplier.
It's still not clear whether Virgin Media fibre broadband is available - Virgin (formerly NTL) use their own cable TV network to deliver the services (rather than the phone lines), and there is not enough bandwidth/capacity in the in-building cables to support broadband.
First impressions is that the service appears to be much more reliable, and certainly significantly faster. If you are a heavy data user, or regularly use "catch-up TV" services like BBC iPlayer, you should definitely see an improvement.