If I read the Southern site correctly then the trains into London Bridge are fast from East Croydon, so at Forest Hill / Honor Oak I think we are back to the Overground (with no doubt a reduced frequency - - also not sure if Overground can run after 6pm?). If you wanted to get to London Bridge by train, there' some going through Lewisham but that's a trek. So not a good picture I'm afraid
Southern got round to finally updating their strike page after I've posted.
As you mentioned, no stopping service to London Bridge, so I'd assume that's due to the Overground running services on strike days which lets Southern concentrate on other lines not served by other providers.
There are trains to Victoria from Crystal Palace and East Croydon if anyone wants to travel to Victoria on strike days, especially on Tuesday when the tube are also on strike and the 185 will be rammed. (East Croydon will require changing at West Croydon for the tram as there is no Thameslink or Southern service from Norwood Junction).
It's worth noting that the Overground including the core section and the Elizabeth line use Network Rail signallers, so they may close early as well due to them being on strike as well.
What Network Rail appear to have done is find staff who either aren't part of the RMT or managers who are trained in using the signal equipment to maintain a daytime service on some lines.