I've dislocated my elbow twice: once when I was about 8 then a year later, so about age 9. Of course at that age I didn't exercise heavily but I still wanted mobility back in my arm, because wearing a sling for months on end is extremely annoying.
The length of time would depend on multiple factors; for example, is your arm in a cast? If so, I had mine on for at least 6 weeks, and wasn't allowed to participate in any strenuous exercise during that time. The first time, once the cast was removed, I had a course of physiotherapy, where I was taught basic yoga-like stretches where I had to put more and more weight onto my arm muscles. Also, I would suggest swimming: it is a seemingly easy workout that forces you to use your arm; afterwards, you will definitely feel the benefit.
As for recovery time, my arm took about a year to fully recover. I could use it within weeks of the cast being removed, but I often suffered - and still do when the weather is very cold - from aching joints and muscle weakness. Then again, I was a lot younger than you are now, and didn't have toned muscles as you would have from lifting weights; your recovery time should be less, I presume.
I wouldn't worry about it too much, or quote me on it, but you may be left with a weakness in the joint: hence why I dislocated my elbow again a year later.
I wouldn't try to rush into any taxing routine because you may end up doing more damage than good; once you start lifting again, I would be careful. If you're sensible, you will know when your arm has had enough, and when to let it rest.
I hope I have been slightly helpful, and that you recover soon!