Traditions nonewithstanding, given memories of Richard III, I could envision Henry VII being strongly advised to send Prince Henry into "exile with the serial numbers filed off" or at least make provisions to exclude him from any position of power should he himself die before his grandson achieves maturity. He'll want to keep him alive as the proverbial spare, but also make sure history wont repeat itself.
Not to mention that with infant mortality being what it is at the time, if the grandson DOES in fact die from an accident or disease, it would also protect Prince Henry from being forever tainted with the suspicion of having murdered his nephew, if he's out of the country at the time.