Getting back on bike after wrist fracture
All but the most gentle rides on smooth surfaces can put quite a lot of stress on your wrists. Listen to your doctors on how long you should rest it because in general there's still healing to be done when the cast comes off. Not only is the bone still fragile but the muscles will be very weak. I was a teenager when I broke my wrist so timings may be different, but the length of recovery after the cast was removed was equal to the length of time it was on. 

Do any physio exercises you're given, which will help you retain strength and movement. Then start in benign conditions - good roads when they're quiet - until your bike handling is back to normal.  When you first get back on, you really don't want to fall off, and wrist-jarring potholes should be avoided (you may not have the grip strength to maintain control, in addition to avoiding the shock). This is likely to be worse on a road bike than on flat bars. Everyone is different and everyone's injuries are different, but if you plan on a cautious approach and take advice from the medical professionals you shouldn't go wrong.