Contrary to popular belief, the bridge has always been fastened at the joints by metal pins, screws or bolts (pedants may still correctly claim it was constructed without the use of nails). However, it is true that the bridge was rebuilt in the early Twentieth Century. Whereas, in older versions of the bridge, the pins/screws fastening the joints were invisible to those walking over the bridge, in this new bridge the joints were fastened by bolts passing right through the joint, the bolt heads being visible to those crossing the bridge. The Queens’ College website suggests: “One may speculate that perhaps the sight of bolt heads where none had been seen before might have given rise to the myth of the failed re-assembly”.
To dispel another myth; the bridge was designed not by Sir Isaac Newton, but by one William Etheridge. Incidentally, I believe the official name of the bridge is not "the Mathematical Bridge", but simply "the Wooden Bridge".