Southern will probably experience far fewer problems than, say, Greater Anglia or First Capital Connect. Southern's trains use the 'third rail' electrification system, which doesn't suffer the problems of overhead cables swinging in the wind which reduces running speeds on lines with overhead electrification.
Interestingly, despite the Mail (which is hardly noted for its accuracy anyway) quoting a time of 0900 for the commencement of Southern services, that company seems not to have fixed a time:
http://www.nationalrail.co.uk/service_disruptions/60171.aspx
News locally just said no dustbins being collected tomorrow either, to avoid empty bins sailing round the streets. Ferries are still running according to plans at the moment, unless it gets to more than 55 knot winds.