Some years ago, I read the Newton biography "Never at Rest" by Richard Westfall. However, I don't currently have access to a copy, so the following is from memory.
According to the biography, Newton has no formal mathematical background to speak of, nor was there really much by way of mathematical background to be had at the time his career began. However, after becoming a student at Cambridge, it took him approximately 18 months, from a standing start, surrounded by a whole lot of nothing (ie. mid-seventeenth century Cambridge) to start making original mathematical discoveries, including the Binomial Theorem, and the early stages of the calculus. Much of this work emerged from things like formal polynomial manipulation.
His background was essentially his own reading of people like Descartes, and Fermat.
My recollection is that the main historical record of this is his notebook (called by him, his "waste" book), which was given to him by his stepfather, Barnabas Smith. But it is unclear to me how historians know what period the notes therein were made.
The upshot is, I'm wondering if my description above is still the standard understanding of the Newton mathematics chronology, or at the least the beginning of the chronology.
The description given above just seems a little improbable. It seems more likely to me that such substantial mathematical advances came at a later period of his life, and with more effort than would be suggested by an 18 month run up from a standing start to original mathematical discoveries.