No one really knows because switching a brain is not possible, at least not now. The brain is connected to the rest of the body by a very large number of nerve cells (neurons). It's easy to cut these, but re-attaching them is not easy. It may never be possible. You would have to attach one cut end of a cell to the cut end of a cell from a different animal. And it would have to be the right cell. The cells would then have to join up and heal together.

The other part of your question is a little easier to answer. The pituitary gland is about the size of a pea and hangs off the bottom of the brain. It makes a hormone that tells certain parts of your body to grow. The cells in these places divide faster. The pituitary gland is also controlled by another gland, called the hypothalamus. Hormones from the pituitary go into the blood and travel all over the body. Not every cell responds to the hormone, only the cells with receptors for the hormone.

Why would it be important for all parts of the body to get the signal to grow at the same time?

The animal would die.

Organs of an animal are integrated into the body of that animal. Your brain could not be removed from your body without killing you, and it couldn't be put into someone else's body, even another human, without killing them, too. There are some organs that can be moved between individuals of the same species, hence the field of organ transplants, but it definitely would not work between species. It might one day be possible I suppose to swap the brains of two humans, but if you did that you would create two new personalities, since brain and body work together. Between species, if the organ being moved was not vital, it just wouldn't work, and if it was vital, then the outcome would be death.

Growth is done by stem cells, cells with the ability to become multiple different kinds of cells, and is further regulated by hormones, and the activity of certain genes. This is a complex field.

It is nearly impossible to transfer brains between species because the correct neuron connections aren't there. Same as if you would try to plug a guitar into a telephone. Many different things must come together correctly for the body to grow--growth hormones from the brain are an important part of this. But, the adult brain no longer releases hormones for limb growth (if you lose an arm it won't re-grow). So the short answer is if you place the brain of one species in the body of another, both species will die. We don't yet have the technology to correctly hook up the brain even within a single species.

One neurosurgeon successfully transplanted one dog's brain into another's body, but as it couldn't be hooked up correctly, it didn't control any body functions, but the brain itself lived. I had no idea that this had been done.