At the beginning of each player's upkeep, that player puts a flood counter on target non-Island land they control of their choice. That land is an Island for as long as it has a flood counter on it.
At the beginning of each end step, if all lands on the battlefield are Islands, remove all flood counters from them.
The land stays an Island until the flood counter is removed, even if Quicksilver Fountain leaves the battlefield.
The timestamp of the land-type-changing ability is set when the triggered ability resolves (so each one has a different timestamp).