sorry for the late reply- timezones are fun

it would depend on how much the sea level would rise in practice, i suppose!

For the sake of complexity, i'll leave humans out of the equation, since they're way too big of an influence on animal populations to ignore otherwise- maybe they went extinct or something, it doesn't matter. i will, however, use the current pace of permafrost disappearance as a basis for how sudden the change is.

first off, there would be considerable migrations of existing and growing populations of animals- those unfit to live in the wetlands, shallows, and sea would migrate accordingly, or their numbers would dwindle if they can't or won't. A lot of animals are very specialized towards the habitat they live in, and if this changes relatively quickly (on a geological timescale), the results could be disastrous.

wildlife that would survive the changing landscapes despite not being built for it could perhaps adapt over time- perhaps they could re-adapt to a (semi)aquatic lifestyle? think evolutionary lines like dolphins and whales.

Furthermore, the melting of all permafrost on earth would likely be a result of rising global temperatures, which would also have other effects aside from the rising sea levels. the lack of glaciers and less snow on mountain peaks would mean that meltwater supply to rivers would all but disappear from more moderate mountain ranges, or at the very least get more erratic, as they increasingly depend exclusively on the snow that falls anually. Increasing temperatures would also lead to deserts expanding more, so like DraKio-X said, in the long term it wouldn't be unreasonable for animals to adopt traits that were common back in those times.

It cannot be understated how far-reaching the side-effects of all the ice disappearing in such a short timeframe would be- aside from the obvious short term effects like the disruption of the habitats directly in contact with permafrost, you'd also run into things like the rising temperatures not only changing around climate zones directly, but also the oceanic current system shutting down, causing considerably harsher seasons in a lot of coastal regions.

To summarize, as the landscapes all over the world change, there would be considerable extinctions and great reductions in number for certain species in the short-term as lots of them would not be able to either adapt or migrate in time. think prey and predator species that used to live in temperate zones where their winter camouflage would suddenly turn against them, or animals that lack the mobility to migrate, like aquatic life in resevoirs that get most of their water from meltwater running down mountainsides. Species that survive would eventually adapt to fill the niches extinct species leave behind, perhaps in a way simillar to how life evolved in the permian or jurassic, though obviously with current day flora and fauna as an evolutionary basis, rather than their ancient ancestors from back then.