I made the switch maybe 5ish years ago when I felt the early signs of RSI. It took a few weeks of regular practice to get up to a somewhat productive speed, then a few more months to reach my previous speed. Now I would say I'm faster than I was with QWERTY. Also, typing "QWERTY" on Dvorak just feels wrong.

I switched when I got my Advantage 2 QD. I used typing trainers for a couple of days to learn the positions and then it took a couple of weeks to get adjusted and not have to think too long.

Before that, I could not touch-type on QWERTY, but learned the Neo layout a couple of years back. I gave up on it because I was using too many other machines then. It felt more natural, but I think what really makes the difference for me is the keyboard. On the Kinesis, it is very obvious which finger to use for each key. I was a lot more confident in the physical mapping within days than after decades on standard staggered layouts. I wish there were ortholinear options for laptops.

Took me about 3 weeks to get to proficient and a month or two to be fluent. Mainly playing on a dvorak typing tutor website while watching TV.

More time than it takes to productively switch to Colemak.

Edit: you can learn the basics with typing games in a few days but then you commit and switch to it full time and your WPM drops to like 10. It takes a while to build up the muscle memory again and it's a tough experience. Like learning to walk again.

Although you already have a number of answers, still I want to provide another. I've been using Dvorak* for over a decade now, and I love it.

Mind you, I never claimed to be a particularly fast typist on any layout, although I have been touch typing for several decades, and on that note I am just as productive on Dvorak as on qwerty -- but at far lesser expense. That is to say, don't switch for the (mythical?) increased typing speed, switch for the ~30%(!) reduction in finger movement and pleasant "rolling" motions you gain.

True, it's not easy to share computers, be it you using somebody else's machine or somebody else using your machine. The latter can easily be addressed by quick layout changes (again, a pain to get right on Windows but a breeze on Linux), while I find that the former doesn't really bother me so much -- curiously, when I can't touch type (such as on my phone), I am hopelessly slow on Dvorak but quite efficient hunting-and-pecking on qwerty (possibly because I retain training from before touch typing?). The most glaring downside is the situations where keys and shortcuts are clearly laid out for qwerty and won't allow you to remap them (case in point: WASD in games); in reality most shortcuts are easy enough to use regardless of layout (side note: I've always been using Ctrl/Shift+Insert/Delete for cut/copy/paste, so the locations of X, C, and V don't affect me -- but modern keyboards with misplaced or missing Insert keys do).

To answer you question: I very specifically switched during the week before New Year's eve, when I could be reasonably sure that things at the office were slow enough to allow me to struggle through an unaccustomed layout. Honestly, the first days were atrocious, but within a week (so ~35 hrs) I had internalized the layout to a reasonable degree, and within a very few weeks (3? I don't really remember) I was up to my usual typing speed.

Bonus: here, have a go and try for yourself: http://gigliwood.com/abcd/lessons/ (no affiliation)

* Specifically, the Norwegian variant of Dvorak, because I need some localized keys. This has been a challenge, but a surmountable one, back on Windows (along with changing the keyboard layout for the Windows login screen, which truly defined the pinnacle of obscurity) -- on Linux it is dead simple to set up and just works.

Not too sure, I switched in highschool when I wasn't being especially productive in the first place. Maybe a couple months for me? I learned it before Colemak was available though, I'd probably recommend that over Dvorak now - it's largely comparable for typing effort metrics and has fairly broad software support, but easier to learn if coming from qwerty.

I found I was back to near my original typing speed in a little over 2 weeks when I switched to Workman layout. I also switched to ortholinear at the same time, which may have had an effect on the time as well.