A thousand years is nothing to an ecosystem. Birds have been migrating across Europe, Asia, and the Americas for hundreds of millions of years, only to get slaughtered in droves by furry shit machines.
It depends on the ecosystem. Pollution famously caused certain moths to shift from being mostly light-coloured to mostly dark-coloured in a matter of years. The removal and reintroduction of wolves in Yellowstone caused observable changes in prey behaviour within a decade or so. Of course longer-lived species like trees take much longer to adapt, but we’re talking about birds, geckos and rodents here.
Edit: Also, most geckos, birds and rodents are r-strategists, meaning they are limited more by food than by predation.
I don’t think the introduction of thousands of F. Catus to any local ecosystem will have anything other than dire consequences.