Uckfield railway station: The sensible rail arrival is Southern’s Uckfield line from London Bridge, via East Croydon and Oxted. Allow roughly 80–90 minutes from London Bridge; the line terminates at Uckfield, so it is straightforward once aboard. Do not assume you can continue by train to Lewes: the railway does not run through.
London Bridge station: Take a Southern service to Uckfield. The journey is about an hour and a half according to the local travel guidance, and is the cleanest public-transport approach from central London. Check the live timetable: this is a branch line, not a turn-up-and-go Tube service.
London Gatwick Airport: There is no direct airport-to-Uckfield train. The practical public-transport route is a train from Gatwick to East Croydon, then a Southern service to Uckfield; allow about 1 hour 30 minutes to 2 hours including the change and roughly £20–£30 depending on ticket type and time. A pre-booked taxi is quicker—around 45–60 minutes in good traffic—but expensive, so it makes sense mainly for groups, luggage or an awkward arrival.