Is it safe to leave exposed black wire from doorbell transformer?
No, it is not safe.

I would turn off power to that circuit at the distribution panel, test the wire with a suitable voltage detecter/meter and insulate the end of the wire until the whole repair can be completed.

If you can't do this and the house is occupied, I would treat it as a serious life-threatening hazard. I would prevent all access to that room, by fitting a lock if necessary.