Have any of the 377/5s that have already moved to Southeastern been repainted yet or had Southeastern logos applied?
Given that the 377s they'll be inheriting from TL have recently had an external clean-up and are now just sporting purple paint, pink doors, 1st class stripes and the new TL logos, I expect Southeastern will just scrub the TL logos and replace them with the blue/white Southeastern logo. They might also plaster blue vinyls over the doors to keep them feeling "closer" to their current brand.
You can be rest-assured that SE will make an attempt, but it'll be cheap-and-cheerful. I'd be incredibly surprised if they send them up to Ilford on the back of the 375/8's and /9's refresh.
They'll almost certainly get the TL logo scrubbed prior to going into passenger service once they're officially on SE's books.
I couldn't tell you anything about software re-flashing, though. Ramsgate almost certainly don't have the capability to flash the 377s, and I don't know if they'll even try. They may just swap the "This train is the Thameslink service" PIS cues with generic ones until they can get something a bit more permanent up.