Here is the definitive answer, for Southern:
Unless staff state otherwise, First Class is in operation on "mainline" services in the London Zones, but not on the vast majority of "Metro" services.
First Class tickets can be purchased on paper or on a lot of smartcards, eg. Southern's "key" cards, but not on Oyster. Oyster is only able to be used in First Class on mainline services at the weekends, provided you are willing to pay the Weekend First upgrade as soon as it is requested of you, which is a blanket £5 cost for one journey on any Southern train. For Oyster, also read contactless, Freedom Passes, Apple Pay etc.
The very observant will have noticed a tiny bit of small print on some of the First Class signage over the last couple of years, which does say that First Class upgrades are only available at weekends.
So... geography...
Metro services are mainly Southern's suburban stopping services, and these trains can be formed of trains with or without First Class areas provided. They are also services which generally spend a great deal (or all) of their journey in the London Zones, but may run outside the zonal areas at times. Routes included are those Southern trains which run to Beckenham Junction, West Croydon, Sutton, Epsom, Dorking, the route between Dorking and Horsham via Ockley, Caterham, Tattenham Corner, Wimbledon, Guildford, Crystal Palace, Streatham Hill, and all stops between London Victoria and East Croydon (mainly peak hours).
First Class is also declassified on some (but not all) of the mainline fast shuttles in the evenings between London Bridge and East Croydon, via Norwood Junction only, and on Southern services which start or terminate at Ashford International, often on the basis that some of these trains are quite short and can be very crowded.
First Class is provided on some Southern services towards Milton Keynes Central, which is mainly enforced north of Watford Junction. All such trains must have a conductor north of Clapham Junction, so they would be the person to ask. These are usually the only Southern services where it could be enforced part-way through the journey. These trains are something of an oddity, in that they are grouped under Metro but run with semi-fast stopping patterns at higher speeds north of Shepherd's Bush.
Trains to/from London Victoria or London Bridge which run via Redhill, Gatwick, Oxted, Three Bridges and/or Haywards Heath generally have First Class in operation throughout the journey, including in the London Zones. These services usually spend considerable time outside London, where people may have longer journeys where an upgrade to First Class is perhaps somewhat more likely to be worth the effort, and are almost always formed of trains with First Class areas.
