I don't think there's one very near Waterloo. I use Andrew Kirby on Grange Road. I would have thought there'd be one nearer to you than that but I can recommend them. They are a bit more expensive than most but worth it.
Andrew Kirby is the closest to Waterloo, and it's a very good vet. There's also Goddards in Kennington Road (there's probably not much in it in terms of distance).
Walking isn't great in Waterloo, the river is always nice and there's quite a few parks around, but getting out to somewhere like Southwark Park or Hyde Park is probably the best thing for a good long walk.
I'm quite fond of Peckham Rye, which isn't too difficult to get to from here, and has toilets, parking and a cafe, all essential aspects of a good walk!
Of course, Riohmond Park is always worth the effort for a day out, your dog will have a good sleep after an afternoon in there :-)