I located one discussion of Brahe's prosthetic nose in some depth, from 1972 in Plastic and Reconstructive Surgery (sorry, paywall), by C. D. Lee; I could not get it at first, but thanks to /u/ProbabilisticWorld (OP) I obtained it. They also confirmed for me that it does mention the affixing of the prosthesis. That essay indicates (p. 333) that Brahe's nose was made via the lost-wax process, replaced with an alloy of copper or gold (it was brass, based on forensic evidence from the 2010 exhumation of Brahe's remains) and then painted to match his skin. The prosthesis was attached with "a glutinous substance which he carried in a small box," so he probably had to reattach it often.

I wanted to find out where this detail came from, however, and whether we get any better information by chasing it down. Lee does not give a direct citation, but I tracked down mentions of this back to chapter 2 of J. L. E. Dreyer's Tycho Brahe: A Picture of Life and Work in the Sixteenth Century (1890), pages 26 and 27, which then cites Pierre Gassendi, likely his 1654 Tychonis Brahe, and refers to a 'some kind of ointment or glutinous composition, which he frequently rubbed on his nose.' We don't know exactly what this substance was, beyond the implication that it was not especially remarkable. Spirit gum didn't exist until the late 19th century, so maybe one of our apparel historians can venture a likely candidate.

As for the commonness of prostheses, I will not speculate because it involves the survivability of various injuries and the prevalence of same, but perhaps one of our Early Modernists can shed more light on that.

edit: fixed link, relinked directly to journal site itself.