I was hired by a Guard unit while I still had 6months left in my degree, so it does happen (I ended up graduating and switching units, but that's neither here nor there.). The worst you can do is not to apply. If they interview you without a degree, chances are they're interested enough to consider hiring you with the time you have left.

The process from hire to starting UPT is usually about 1yr. Here's my timeline:

Hire Jun 2010
Brooks Jul 2010
Guard approved Packet Oct 2010
LASIK Waiver granted Nov 2010
AMS Jan 2011
SERE March 2011
IFS Apr 2011
UPT Jul 2011

YMMV.