Assuming we could do a human brain trransplant and it wouldn't damage too much the recipient body or the donated brain:
The body that is currently called "your body", when it woke up, would wake up with Stephen Hawking's (or whoever's) thoughts and memories, though he may be in different "moods" than he normally is in because now he would also have your endocrine system (your glands), which would affect his brain function.
That's because our best model of how mental activity occurs is it's the brain that stores style of thinking, skills, memories and other types of mental content. This is in contrast, to, say, Aristotle's view that the brain was merely an organ for cooling the blood. Furthermore, specific brains--in terms of structure and function--give rise to specific sets of mental content.
What's the evidence for the idea that the brain is the location of mental function and that the structure and functionality of a particular brain dictates the mental content generated by that brain? A great deal indeed, but here is a smattering of evidence:
Damage to the brain causes obvious changes to mental function. One good examples of this is stroke, in which specific areas of the brain are destroyed. This can leave a person with a very specific mental deficit, such as the inability to recognize specific classes of objects, faces, or words. But there are many other ways for the brain to be changed, and along with it, personality or thinking. Tumors may have been responsible for sexually deviant behavior. On the positive side, a traumatic brain injury may have turned a man into a math genius.
Temporary changes to brain biochemistry, such as alcohol intoxication, hypoglycemia, hypercapnia (too high carbon dioxide), etc., all cause obvious and usually reliably reversible changes in mental function.
There are essentially unending examples, either in humans or animal models, that show that brains underlie mental function and that specific brains give rise to specific minds (for some non-spiritual definition of "a mind").