Interesting. 16.22 explicitly says that when you build a "new bridge", it is marked as controlled by you.
Good question. A blown bridge is controlled in the same way as an intact one (16.20). It follows that if you repair a blown bridge that is controlled by the Allies, it remains under Allied control.
Obviously, it should be easy for you to take control after repair, since no Allied unit can be within one hex in order for you to repair it in the first place.
I wonder whether it's an oversight that 16.21 doesn't say the same. The only evident difference between "repairing a new bridge" and "building a new bridge" is whether that hexside has roads or not. The actual physical work done by Bridge Engineers seems similar in both cases, and they're surely doing construction work on both sides of the hexside...
But indeed a literal interpretation of the rules seems to imply that if you "repair" a "destroyed" Allied-controlled bridge, then it continues to stay Allied-controlled...
That seems very weird to me, and it's hard to imagine the reality of German bridge engineers rebuilding a bridge and it still being under Allied control when they finish their work. I wonder if that's really the intent...
A counter-argument to the literal interpretation of 16.21 is that a a bridge is destroyed, then it no longer exists, so the concept of who controls that non-existent thing is meaningless. I.e. a non-existent bridge can't be controlled by anyone.
But a counter-counter-argument is that a destroyed bridge (e.g. smoking ruins) is distinguishably different from a non-existent bridge (e.g. nothing at all).