How to fix old BX wires
As I understand it, normal cloth is not electrically conductive, so it shouldn't be a significant problem if some of it touches the conductor. My bigger worry would be a failure of insulation leading to a short-circuit in the cable. Applying the heatshrink will have inevitablly involved moving the wires around increasing the chances of cracking.

I would be very wary of touching the installation any more than you absoloutely have to before you are able to re-wire it.