I will provide some thorough personal reflections.
To me it seems like spoken languages are a layer on top of thought, in the same way that musical notes as formal symbols C D E etc are a layer on top of the momentary musical experience. The connection between the two layers can be stronger with people who have perfect pitch or who compose.
My mother tongue is Swedish. This means that Swedish is an interface to my thoughts. But I usually don't use Swedish at all when I do my work (which is math). The reason for this is that English provides both an alternative interface to my thoughts, and an interface to the literature.
Similarly, it is common for kids internationally to use English phrases from video games. The use of domain specific words translated to their mother tongue would feel unnatural to them because it would be a detour.
Extrapolating from this, a polyglot could have different uses for different languages. If I am to be careful, I wouldn't say that he "thinks in those languages" (or in his mother tongue for that matter), because this insinuates that all thought takes place in a spoken language. I would rather say that he habitually uses an appropriate language as a medium between his innate, strong, intuitive thinking abilities and the task currently at hand.
But to avoid belittling the value of language in thinking, I should acknowledge that language can play a great role in structuring, guiding, focusing and spurring one's intuitive thoughts even when not speaking. It is then relevant to note, that different languages could in principle (and probably do) play this role differently for an individual.
The next question is whether a polyglot who "thinks in several languages", can use all those languages at once. According to the principle that we cannot keep many things in mind at the same time, it seems that the answer should be "no". It would simply cause too much interference. It may even typically be too much to use two languages at once. Running somewhat counter to this hypothesis is the fact (citation needed) that some excellent composers can imagine multiple voices of music in parallel. By analogy some exceptional people should be able to use several languages at once. But I think it is not controversial to claim that this ability would be something out of the ordinary.
Some other specific questions are whether a polyglot would use different languages to think about the same topic on different occasions, and whether he would even switch between languages repeatedly while refocusing his thoughts on a particular subject. As for the first question, I think the language he last used externally, as in speech, is more likely to be used, than if he hadn't used it recently. As for the second question, I see two relevant trails for investigation.
Firstly, multilingual people can frequently switch back and forth between languages when talking to each other. It is not clear to me that the reasons for this should carry over to internal thinking, but maybe they do. Secondly, you could think of a mathematician or a modern physicist as a technical polyglot, for the mathematician uses geometric diagrams at one moment, only to write algebraic equations one minute later, and (although my knowledge of this is limited) a physicist may think of a situation with the vocabulary of particles in one instant, while switching to waves in the next. (I won't mention the practices of computer science and biology, which are more about layers of abstraction.) These people systematically use different languages on the same problem, in order to extract (or store) as much knowledge as possible, but their languages are technical and their practice is disciplined, so this does not account for the generality of the question as one about natural, common language.