Google Hangouts doesn't work in the default browser on Ubuntu Touch OTA10.1. It just gives a message "It appears as if you're using an old or uncommon browser that doesn't support common standards".
I also tried it with Firefox, but it claims to need a plugin, which I can of course not install.
The Ubuntu Hangups app doesn't support calls (video or voice), so it doesn't help either.
It turns out that this is a known issue, and supposedly fixed in the rc-proposed channel (meaning that the fix will be available in OTA-11 in a few weeks). See https://plus.google.com/+OliverGrawert/posts/aZi13q1bcWw and https://plus.google.com/u/0/+MikeKelly87/posts/32EUgtorgWJ
I have tried it using the default browser after switching the tablet to rc-proposed and can confirm that it does work (albeit with some problems selecting the camera): https://plus.google.com/+JensGrivolla/posts/JyrN4sJSqst