How to make bridge in repairs impassable for routing?

Keep in mind that routing engines take some time to update their data and re-calculate their routing graph. Your changes are not immediately reflected by routers although the renderer already shows them.

Also I think the current tagging isn't quite clear. There is an access=no tag but there is also a more specific foot=yes tag. I think some routers still allow foot routing along highway=construction if foot=yes is present. The same may apply to bicycle routing. Therefore I would remove the foot=yes and bicycle=yes tags and re-add them as soon as the construction has finished.

CharlieHotel, If a construction only takes a few days don't tag it at all, its to short to be practical. A router should start to recalculate as soon as you leave the original signed route. So even when you did not notice the official D tour signs "follow 3/D" until your back on the old route or way, the router will guide you perfectly to the old / original destination if the road/bridge ahead is signed as closed.

This bridge was damaged in a flood on 29 October 2015 and is in the process of being replaced with a new bridge. To stop the routing engines I found I had to explicitly set each mode of transport to no. I.e. access=no, foot=no, bicycle=no, horse=no, motor_vehicle=no, cycleway=no, etc. This caused the section of road beyond the bridge to become a routing island in overpass turbo. Even though an alternative access route for residents and service vehicles existed across private land because that alternative route had private access restrictions on it, the isolated road remained a routing island.

Perhaps you need to explicitly tag foot=no to prevent the pedestrian routing.