The leak has finally been resolved, and the upstairs owner has said he will repair the damage to our apartment. I have not seen anything in writing from him directly, but his real estate agent (it was rented out) has said in writing that he has agreed to the repairs.
I would send them a bill for the repairs with a fairly short “pay by” date. If they don’t pay, you could consider starting proceedings at a small claims court.
That said, when owners are trying to sell, they don’t want any loose ends lying around that may affect the sale, so they are likely to clear the debt rather than have awkward questions being asked.