From a camera perspective the Nikon D5000 and Canon 600D are both perfectly usable and powerful enough for wildlife photography. However, neither one has a kit lens that would be useful for wildlife, especially birds.
As a very general rule of thumb, 300mm is the minimum for larger game, 400mm for larger birds, and you can never have enough focal length for the smaller birds. Often you'll find pros with a 600mm lens with a teleconverter to get even more length. With lenses of this length you have to start thinking of a good support structure like a good tripod with a good head.
However, there are good budget means to get adequate length for birding such as the Sigma 150-500 and 50-500 lens and the Tamron 200-500 lens. They are available for several camera manufacturers including Nikon and Canon. They give good performance for the price but they have limitations, too.
Another popular way of doing birding where you get a lot of length is with digiscoping. In this technique you use a spotting scope, which typically has more of a telescope like power, and add a camera to the end. Some rigs cost more than expensive lenses but with better focal length but some are more affordable.
Lastly, consider how much work you're willing to do for bird shots. You can get by with shorter lenses if you're willing to put in the work to use blinds and other techniques to get closer. The best birding pictures are those where the photographer has worked their way into a great position that doesn't interfere with the birds, is in good light, and a well chosen background. Also learning bird behavior is essential to good bird photography. The more you learn the more you can work around equipment limitations.