The larger wildlife of the Meadows & Links (party jokes apart) is mostly:
(a) birds, about which I know next to nothing, and
(b) squirrels, of which I like to take countless blurred pics.

For the blurred pics, check out Squirrels and Birds.

For the local mysterious owl, check out The Argyle Owl.

There are five stately Unicorns who guard the park. They always hold still for you while you take their photo.

Try "Edinburgh Meadows" in Flickr for a chance of spotting pics of the birdlife (but you'll find that most people like to photograph the human life and the cherry blossom).

The park has a respectable if not large population of smaller wildlife such as insects (see Bees and Ladybirds), and it feeds a good population of birds and squirrels (plus any mice too cautious to take up residence in the neighbouring flats). I've yet however to see much in the way of larger animals apart from the ubiquitous adorable domestic dogs.