You can normally negotiate your way through, though two instances where I failed: one in the Teign Valley, and as I gingerly proceeded towards one of the works supervisors, he told me in no uncertain terms to remove myself from the works area. To be fair, there was a lot of heavy machinery involved in stopping the road falling into the Teign.
The other one was in France, where there was a sign saying "Route fermée, même cyclistes" ("Road closed, even to cyclists"), and after a friendly driver had directed us via a rather circuitous detour, when we looked down on where the road had formerly been (it was trying to fall down into the valley), we were grateful for the clear signage, and the detour.
Otherwise, generally there's a way around these things.