Prosthetic leg

Thoughts:
1- Pain could be a huge issue with an amputation. General anesthetic was pretty much unheard of until the late 19th/early 20th centuries, so when they cut the leg off, he would probably have only had a shot of whiskey. Ouch.
2- Infection, too - they didn't have penicillin until the 20th century.
3- Even with the artificial leg, it's quite possible that they will still have to deal with back pain from the imbalance between their legs.
4- Here's an interesting article that discusses how the location of the amputation can actually affect long term function negatively:
http://www.nytimes.com/2012/05/15/h...e-unthinkable-choice.html?pagewanted=all&_r=0

Back to your original question... Currently, I think it takes several months for an amputee to get used to the leg, and I would imagine that in the 19th century, it would be a couple of years at least. BUT a lot of that would depend on your story and your character. Is he already a confident person? Is he devastated by the amputation or eager to work hard to get back to his earlier function? Is he hoping to be able to just walk or does he want to resume mountain climbing? Plus what kind of medical care did he get immediately after the amputation and during the recovery? What kind of support does he have?

A lot of it depends on the fitness level of the person before the amputation. Also, it is above the knee or below the knee? Below the knee amputations are typically easier to adapt to.

I wouldn't discount people's hardiness, especially back in the day. It wouldn't be one shot of whiskey necessarily, maybe a whole bottle. Maybe none. Folks were pretty dang tough back then.

I have no specific knowledge of recovery after amputation back in the 1800s, but I work with amputees at present. The Anglesey prosthesis appears to be an above knee prosthesis, which is more difficult to walk with due to the fact that the user has also lost his knee joint.

Keep in mind that there will be a considerable healing time after the injury, in which the residual limb will have swelling and the actual wound closure which must heal well enough to bear weight in a socket. Those sockets (wood and leather) wouldn't have the modern cushioning (gels and such) nor the exact fit that today's prosthetic sockets have (today's are custom molded). Depending on the scope of the injury and the person's vascular supply and muscle flap, it could be months before the limb is ready.

A younger, fit person, such as a soldier, generally has a shorter adjustment period to using a prosthesis, due to having better balance/coordination/strength, etc. than, say, an elderly person who has lost a limb due to complications of diabetes or something like that. If the prosthesis fits well, many people can stand and walk pretty quickly after donning the prosthesis. Of course, mobility will be improved and fine-tuned, including not needing a walker or parallel bars, and working towards uneven surfaces, steps and so forth, over time.

Here's a YouTube video of a young woman walking for the FIRST time with an above knee prosthesis. See how quickly she gets it?

http://www.youtube.com/watch?v=3mrKwui3S3k

I don't know if you mean only the time to actually learn to use the prosthesis, or the time from injury to when a person would be able to walk with a prosthesis. Those two questions have very different answers, due to the healing as I mentioned. And depending on the time course you want, you can give the character more obstacles to overcome, or fewer. Good luck!