According to Lewisham's own web site

In inner London boroughs all roads, pavements, footpaths etc. are defined as Public Highway, Highway or Private. It appears that this pathway falls under the category "Private" as it does not appear on the Highway Register which means that it is the responsibility of the landowner. Landowners can limit access to Private roads and paths in any way they feel appropriate (i.e. gating etc), and as the owner of the land they are responsible for maintaining them.