With the camera's WiFi enabled, you can establish a connection to the phone. It does not require externally available WiFi.
There are no provisions to utilize a data transfer to the mobile in this manner with an HDMI cable.
Go into the WiFi settings on both the camera and phone and make sure they are both on the same network. I ran into this problem when my phone was on a 5ghz WiFi network at home and my camera was only capable of being on my 2.4ghz WiFi network. Once I switched my phone over to the same 2.4ghz network they connected.