How long does it take for a person's body to become accustomed to exercising in the heat?

It really depends on the person. If you've lived somewhere cold for an extremely long time, and if you only exercise every once in a while, your looking at months. If you exercise everyday, however; it may be a few weeks. But like I said, it depends on the person. I'm one of those people who refuses to exercise outside during the winter because I've never grown accustomed to the cold. (I do live in Texas so the heat is hot and the cold is tepid to other people XD, so take my advice on the heat part.)

My advice on how to become accustomed to it is to spend a lot of time outdoors. Not only is it healthy, but it will also speed up the process of getting used to the warmer climates.