Thanks for the help. Great response.
The port checking tool is very useful indeed:
http://www.canyouseeme.org/...
I installed the socketserver on the device at home so that it would listen for connections. The port checking tool connected immediately with the device at home proving that the port forwarding, DMZ, Firewall are OK. BTW, on the router, I reduced the number of open forwarded TCP ports to one, as suggested!
Moving on, the mobile phone has the same socket server code with the same port open. It is on mobile Wifi only.
ISSUE FOUND: The port checking tool cannot connect to socket on the mobile phone.
Are there any things that need to be done on the phone which might be blocking the port? System settings I've missed? Firewall?
Now I know where to look...the battle continues...