I touched typed on Qwerty before learning Dvorak. It took me about 3 weeks to get up to speed. It was extremely frustrating initially, but I'm delighted with the results (ten years down the line).

It took me about a month to be comfortable with Dvorak. It's probably going to depend on how much you are willing to commit to Dvorak. You will learn much faster if you can avoid the temptation to switch back to Qwerty when you need to type something quickly. Typing a lot will obviously help as well.

Initially my Qwerty skill dropped way off, but two months later I was able to switch between the two formats with little effort.