I don't know the exact legal position of the subway but I believe it is an established right of way. It is owned by National Rail but the cleaning and lighting is carried out by Lewisham Council. The renovation work has been commissioned and partially paid for by Lewisham Council (with the majority of the cost coming from the contribution from Berkeley Homes when building City Walk).
I assume that because it is a public right of way there was the need for a notice of closure under the Road Traffic Regulation Act, even though it is not owned by the council.