No Random Timer?

[HyoMinPark]

It is put on a timer. The plot will still display as empty, though. To check if the house is purchasable or still on a timer, you need to interact with the placard.

[MizArai]

Also, if your FC already owns a house, you can by-pass the random timer by relocating to a new plot. So, if there's an open large plot, you could just relocate your FC house there (assuming you're not over the limit for the amount of stuff that would need to be placed in storage from relocating).