IMO, it depends a bit. Are we talking talent with regards to learning techniques (motor behaviour, if you will) or with strength, endurance, etc?
With technique, there has been suggested something called "specific motor abilities" which are supposed to be pretty much genetic, and won't respond much to training, however, no one can answer where these actually are. When you are born (or before that, really) your nervous system is very undeveloped, you just have the basic structure (which is probably genetically determined) but after that your CNS starts to develop based on both genetics and envoronment. It's a pretty well known theory that we learn by the CNS changing. So if I for example kick a fotball a certain way many times and it works great, there will be changes in my CNS, making that specific technique more likely to happen the next time I want to kick a ball. Now, since these things develop based moslty on what you do, then environment and behaviour would have big impacts here. Now, could some people's CNS respond quicker? If that's the case, they could learn a given technique quicker (but that would go for both a correct and an incorrect technique) however, if you ask anyone where this would come from, they probably wouldn't be able to answer.
With regards to strength/endurance. A lot of people use twin studies to argue that genetics play a big role. They refer to studies in which twin pairs who have usually been separated at birth (thus had different environments) respond very similar to the same training stimuls. Now, this would support that genetics play a part in who responds to what kind of training, but sometimes they further use this to say that genetics plays a big role in determining the amount of response to training in general. What they forget, however, is that their study only tests one kind of training stimuls. Maybe the twin pairs who didn't respond so well to that specific stimuls would respond better to another type of training? Some people have a naturally high percentage of fast twitch fibers, and will have a good potential for getting good at "fast" sports, others might have a high percentage of slow twitch and will have a good potential for getting good at "slow" sports. (bear in mind, that fiber type can probably be changed to some degree if you train)
But if you look at a lot of BBers PLers, etc, you will see that they were small and weak when they were kids. A lot of people think that the BBers and PLers who are good now must have been the strong kids in kindergarden aswell, but it's not always so. Now, you could think of this two ways. Either, they were small, got motivated to get bigger and stronger, and they did so, blowing many people with "good genetics" out of the water. Or you could think that they really had good genes, but were small because of bad nurture (not enough food, training, etc). Regardless, both training and nutrutre probably play a role in training response.
With regards to CNS development, now a days people pretty much say it's nature and nurture. Try to google "probabilistic epigenesis". Gilbert Gottlieb's theory of probabilistic epigenesis is interesting to read. You should be able to find stuff about it if you just google.