Not being familiar with the area, I can't work out where "the hill" would be, but the path as mapped at https://www.openstreetmap.org/way/32604306 seems to correspond to FP133 at https://www.bexley.gov.uk/sites/bexley-cms/files/2017-11/Definitive-map-for-South-of-borough.pdf and that probably takes legal precedence over what OS shows, unless and until Bexley ROW team can find historical evidence of a footpath where OS shows it.