Hello, I'm Dr. Kara. I have over 20 years of experience as a veterinarian and I'd like to help with your concerns about Allie's tail.
This is actually more common then you would think. Cats get their tails caught, panic and pull causing trauma or they get their tail accidently slammed in a door and because the tail has little soft tissue to protect the bone, damage can be severe.
Unfortunately broken tails
that have bled and have obvious bone exposure don't heal well with home treatment only. I think that the white stringy material you are seeing is ligamentous fragments that once connected her tail vertebrae. Her fur won't grow back because the skin is missing from that area of her tail.
Tails are easily damaged, are very painful when there is any sort of trauma, and heal poorly because there isn't much soft tissue on the tail around the nerves, blood vessels and bones to help nourish the broken pieces. She will likely have to have the tip amputated, and possibly a few inches further up to take into account any soft tissue that has been crushed, depending upon how much damage is done. She definitely should be on antibiotics and pain medication.
If she is eating and drinking normally tonight you can wait to have her seen by her veterinarian in the morning if they are unavailable now. In the meantime she should be kept very quiet and indoors so her tail wound doesn't become further contaminated.
In some cases if her tail was pulled when it was traumatized she can have nerve damage to her tail (which may be why he isn't showing much pain). In some cases if the pull was hard enough or close enough to her pelvis she may have nerve damage to her rear legs, anus and bladder so watch her closely tonight as your veterinarian will want to know if her eliminations remain normal and she is able to get around normally. Sometimes the nerves are just stretched, but in some cases they tear and when that happens the damage can be permanent. Cats may not show the full extent of their injury for several days as secondary inflammation and nerve death occur.
Best of luck with your girl, let me know if you have any further questions.