If New London falls, a new line will be drawn in Cambridge, if Cambridge falls, Sprague and even Gran Canaria next, if they fall, then it will go to ground, but Bretonia will cease to be a threat to Gallia in the wider picture. A puppet state may emerge, and Bretonian ships may be facing Liberty as a new Bretonian faction that colludes with Gallia. As for Liberty, if Gallia smashes New London and the liberty forces with it, tell me, why would Gallia stop with Bretonia? Liberty remains the biggest threat, with Kusari down, Bretonia down, and Rheinland not as big an issue. We would have a push into New York for a final showdown, potentially with Bretonian ships on both sides.
To answer other questions, the Queen will go into exile, not abdicate, look at WW2 British contingency plans for an invasion of the Isles, they will continue to run until there is nowhere to run to. Maybe the acting government in Bretonia will be a principality, or a vassal state, seeing as Gallia is an Absolute monarchy.
Corporations can decide their own fate, would kind of be a chore to split them in two, probably makes sense for them to become Gallia loyalist.
Corsairs have no role to play in occupied Bretonia, why would Gallia want a bunch of pirates overseeing what they would see as their territory, Corsairs should turn hostile to Gallia, its not like Gallia will give them Cambridge.