Don't go anywhere near Virgin. They have over-subscribed areas of SE23. You would be grateful for 11Mbps at peak times on Virgin 'superfast'. I have just had to fight them tooth and nail to be released from my contract.
I couldn't get fast broadband from BT, and apparently live too close to the exchange to even get connected to the network (weirdly, but something to do with there being no cabinet between us and the exchange, requiring a dedicated line)
I use Virgin and generally have no problem. Have good 60mb (down) most of the time, though only 3mb up.
The reason I suggest Virgin is the doesn't require co-operation with OpenReach therefore no changes to cabinet infrastructure required which I cant see happening any time soon.
I'm spreading my bets anyway, so signing up to both is advisable as if one doesn't acknowledge demand, the other one might.
And for the particularly troublesome areas, see attached (although its broader than this, this is the widest I could screen grab with test scores)