Maybe we can remove WebIOPi from the equation temporarily.
You can temporarily install another web server on port 80 for testing. It will not interfere in anyway with WebIOPi.
I recommend lighttpd for testing.
sudo apt-get install lighttpd
After it finishes, the lighttpd web server should be running, listening on port 80.
You can enter your Pi's IP address into the URL bar of a browser on the same LAN and it should show you the default placeholder web page from lighttpd.
Assuming that works, then run the Weaved installer and install a Weaved service for http on port 80.
Next, connect your phone to cellular, disconnect it from Wi-Fi, start the Weaved app and connect to the lighttpd service. It should open up the same default page. This would indicate that Weaved is working for http connections. If not, we'll have to dig into it further because it really should work!
To remove lighttpd after testing, you should run:
sudo apt-get remove lighttpd
although in general it does not cause any harm to leave it there.