Ensure both CB and CB Gallery are up to date. CB includes jQuery Mobile and we provide TouchPunch already so there should be no issues with touch events passing through. I could not confirm using Android. I do not have an iOS device to test, sorry. It is jQuery based so if you've fatal JS errors present on the page you likely will have issues uploading.

Works fine locally and on demo site. Only works fine on your site with double click. You're using JCH Optimizer to cache the headers. You need to clear that cache manually by editing the JCH Optimizer system plugin and clearing from there everytime you install something. It does not clear JCH Optimizer when clearing in Joomla cache. Once done see if issue is resolved. If issue persists it's probably the ajax component loading behavior you're using breaking JS bindings.

As changing templates fixes it I suggest digging into your templates parameters and begin toggling functionality off until the issue resolves. CSS can't block a touch event unless for some reason there's an element sitting on top of the page blocking events, but I see no such element. Disable JCH Optimizer entirely and retry to be sure that's not what's causing it. Aggressive caching can easily break jQuery bindings and to listen to touch events in jQuery it requires jQuery mobile as well as TouchPunch, which if they can't bind properly they won't work. If the only thing that fixes it is switching Joomla templates then contact your template developer as I don't know what else to suggest as I've no idea what they're doing in their template that could be breaking touch events, sorry.