It's the pilot's responsibility to ensure his nominated alternate is suitable and available; as has been said, the FPL does NOT go to the alternate thus the alternate can have no responsibility to inform the pilot if it's not available. The RAF method of 'booking' diversions either en masse or for a particular aircraft has a lot of advantages, but it's very time consuming and not manpower efficient.
At my airfield, we sometimes have to delve into the inbound FPL for aircraft which will arrive after we close in order to suggest and warn a suitable diversion.