Runs great in Chrome for me. I have the android app and this html5 version runs really well... after playing the first couple of levels, it runs just a well as the android game.
Tried loading the game in android browser, no go... also tried loading it in the dophin browser within android... no go (both just site at 00%).
Was curious what they were using for playing audio so I had a look at the code. They are using SoundManager2 (http://schillmania.com/projects/soundmanager2/) which will play audio using either html5 or flash (it's great, have used it on a few projects in the past)... I believe it dynamically detects what's available. Looking at the code though, I see they (MS/Zepto Lab) are only using html5 audio on IE9 or greater, and flash for everything else... if you want to force html5 audio, use this link:
http://www.cuttherope.ie/?html5audio=true
This might fix issues people are having for browsers that don't support (or block) flash. Haven't tried in an iOS browser, but that link might increases your chances of it working there :)
Using that link I was able to get it to load in browser on Android and started the game, but couldn't cut the rope, likely cause it's not built to handle touch events. Too bad.