For that budget, you'll be looking at an APS-C sensor, which is smaller than the full frame cameras.
You'll often hear people quoting full frame focal lengths, which can be confusing for an APS-C user. So a 50mm lens on a full frame camera and a 35mm lens on an APS-C camera will give the same field of view.
I'm going to use APS-C measurements to make things less confusing.
Your standard zoom lens that comes with most cameras will be an 18-55mm zoom. That's fine for most standard stuff. Pictures of people and most scenes. It will be fairly useless for wildlife though. Some APS-C cameras do longer zooms like 18-200mm but it's worth mentioning that the more zoom range something has, the more compromises there are likely to be with image quality. Basically an 18-55mm and a 55-200mm lens will typically get better results than a single lens that covers the whole zoom range. But obviously then you have to carry two lenses.
For wildlife, you really need as long a lens as you can afford. Nikon do a 70-300mm lens for example, but it's probably out of your budget (although don't be afraid to buy second hand - lenses hold their value much better than cameras).
A wide angle lens (10-18mm) is useful for the ultra-wide landscape shots, but you can still shoot perfectly good landscapes without one.
So once you'd decided on the focal lengths you need, you need to think about the f-stops. Basically the lower the number the better. Lower numbers will perform better in low light as well as creating more background blur if that's what you're into. You might notice that a lot of cheaper lenses will have two numbers (e.g. f3.5-5.6). This is because the lens can't let as much light in when it's zoomed in as it can when it's zoomed out. You can get lenses that don't do this, but they're expensive.
But your other option for low light is prime lenses (i.e. fixed focal lengths with no zoom). These lenses will typically have a much lower f-stop and perform much better in low light as well as generally offering sharper images. They're the ones everyone uses for great portraits because they also have much more background blur. I'd recommend getting a 35mm f1.8 simply because you can probably pick up a second hand one for next to nothing. Primes are the ones that take the kind of pictures that people usually think of when they think about a proper camera.
As for actual cameras, I've got no idea. It's been a while since I looked. But your big question would be mirrorless or DSLR. I suspect you'd get better value for money with a DSLR.
Oh, I forgot to say that it obviously depends what wildlife you want to take a photo of. But most wildlife is small and far away. For things that are small and close, a budget option would be to buy extension tubes for your lens, which will turn any lens into a macro lens for about £20. Great for pictures of insects and stuff.