You need to switch the wires from the bridge pickup over. Black is normally the ground wire on 2-wire pickups. But the bridge pickup on a Telecaster normally has a baseplate which is connected to the ground wire, and if the bridge is grounded, then the mounting screws normally also make a connection to the baseplate and also ground it. If you've got the black wire going to the switch, then the baseplate is connected to that, and then the signal gets grounded through the bridge. In the middle position the bridge ground also grounds the neck signal, so that's why only the neck position works.

But you will probably find that the mixed position is then very thin and honky sounding. This is because the black wire from the neck pickup needs to be the ground and the white go to the switch. Reversing them reverses the signal polarity with respect to the bridge pickup, resulting in a lot of low frequency signal cancellation. It's an interesting sound, but not as useable as the noemal mixed position is.

So you had the pickups correctly wired before, but you probably had a ground wire touching a signal connection somewhere. You've screened the control cavity, and it is fairly easy for this to happen. Always a good idea to put insulating tape over the copper at any place a potentiometer or switch or jack connection could touch the copper.

Hopefully that should get you working.