The liklihood of re-attachment is important.  I would say see your doctor, and short term try to re-position the nail, wrap it in guaze and get a gel toe protector from the drug store.  You may want to try hydrogene peroxide and neosporin on a regular basis to avoid infection. 

The healing time depends on the level of injury, interventions and so on.  You could probably run in a couple of days if it is stabilized and wrapped, but it would be best to get a professional medical opinion, especially if there is any additional bleeding or pain.  If you totally lose the toenail, that is a different story, so just be cautious.

Wishing you all the best and a speedy recovery!

I'm going to go against the grain here and say, keep it clean (soap and water, a little neosporin if you can get it where it would be helpful), tape it down nice and tight, and get running.  Same thing happened to me right in the middle of my first 5K training program.  Foot Dr. said, if it's coming off, it's coming off, you're not going to fix that.  It hurt like a bugger for 4 or 5 days, I just used some really good medical tape to keep it from getting pulled up again accidentally, and then it stopped hurting altogether.  About 10 days later, the nail just fell off altogether.  Ugly?  You betcha!  Still is. 

Good luck!