I’d be looking to get couple of dehumidifiers running tonight in the flat. In the kitchen, remove the kick boards and put a fan or two in there to force the air circulation. Carpet without underlay might dry out OK, with felt or rubber crumb, I’d be looking to lift the carpet straight away, probably removing the underlay, dry the floor but laying the carpet flat again to dry, crossing my fingers that it’s reusable.
Take a bottle of wine and some flowers round to the downstairs neighbours, and offer to drop light fixtures/roses down and supply them with plug in lighting until it dries out(flush spotlights?) and refit in a day or two. Offering to get their system checked again soon by a good sparky would be a nice touch too.
I’d be making sure I had keys too, so no-one needs to go breaking in anywhere, anytime. As for who pays? I’d not be worrying about that yet, but would be looking mostly at getting a new agent if they don’t sort it. How much is your LL insurance excess? Mine is £250, hence my first question about how bad the damage is. Also getting them involved might slow things down if you have a tenant due to move in.