How long does it usually take someone to recover from leg amputation?

Months. It takes months before a stump wound heals and the swelling is fully gone. This about the time it also takes before you can try to be fitted for a prosthetic device.
Depends. The recovery from an amputation depends on a number of different variables. Generally for an uncomplicated case this requires a few days in the hospital and then rehab. The real prosthetic training will probably not start for 6-8 weeks or until the sound has completely healed.