I broke my tailbone in a car accident when I was pregnant with my daughter (yeah, I don't understand either how that happens when your butt is against a cushy seat - but apparently it's a quite common injury with pregnant women and car accidents!).
It was about a year and a half before I had absolutely no pain - it was about 3 months before I could sit in a cushy chair comfortably, 9 months before I could sit on a hard chair (but not for too long) and about a year/year and a half before I had no more pain at all. But again - I injured it while I was pregnant and had to push a baby past that tailbone 3 months post-injury so my healing time was definitely affected by that.
Honestly I would not bother going to the doctor. I was told they wouldn't even do an x-ray (even after I wasn't pregnant anymore) because regardless of what they found they wouldn't be able to do anything about it. They said the only way they will even do an x-ray is if the pain shows little improvement at a year post injury.
The key is avoid pressure on it whenever possible.. if you have to sit, lean forward so you're not putting pressure on it, get a donut cushion for sure. The best thing to buy actually is one of those seat cushions with the cutout at the spine. You will feel like a bit of an idiot, but seriously, carry it wherever you go where you will have to sit. You will heal a LOT faster.