I've had the same problem, the speed options in neither the downloaded nor browser version seem to work. The example text in the menu does change speed but the actual dialogue does not.
While clicking does speed it up somewhat, having to click mutiple times if a piece of text contains more than 1 sentence is rather annoying. Especially since you have to wait for the character movements to finish first before you can click to show text.
The game seems nice, so I hope it gets fixed.