Is this a different process to how a building becomes a Listed Building?
I’ve never heard of the ‘Local List’, or that nominations were being sought. Anyone else?
They’re different, with a different set of rules for the building when it’s listed but with a similar principle behind them. Proper listing is done by Historic England (formerly English Heritage). Local listing is where the Council can decide that a building should have some additional recognition of merit. Anyone can nominate them and the Conservation planning people will check them against the criteria. When I worked for local government (not Lewisham), there used to be occasional calls for more nominations.
More here: https://historicengland.org.uk/advice/hpg/has/locallylistedhas/
And Lewisham’s own stuff is here: https://lewisham.gov.uk/myservices/planning/conservation/locally-listed-buildings