Hi sassy - late in as usual. I'm so sorry to hear about your cat.
Our (now deceased) cat Smudge lived for years with 'half a tail'. She was walking through the hall into the kitchen & the door slammed shut on it. My husband quickly opened the door for her, but she ran upstairs, blood everywhere, & hid under the spare bed. We eventually coaxed her out & her tail socket was hanging limp, so my husband & me rushed her to the vet. They operated within an hour or so, cutting off & removing the broken part of the bone & the rest of the tail socket. They then put a few stitches in the end.
Although it was very traumatic for her (& us), she recovered very quickly & was back out in the garden, chasing the squirrels & birds before we knew it.
I hope your cat will be okay & lets, know how it goes at the vet...