Those outputs look good. The W1f means "walk 1 mode, forward" for example.
So, let's review:
1) The robot hardware works because demo mode works. This rules out electrical or motor problems on the robot.
2) The gamepad buttons are working, and the gamepad is trying to transmit the protocol to the bluetooth module.
3) The bluetooth modules are blinking in the correct way to indicate they are paired and they are talking to each other.
4) The configuration screen proved that you are setting both uarts to 38400 baud, which is what the gamepad and robot want on the connection to the bluetooth modules.
The only thing left is the connection between the gamepad nano and gamepad hc05, and the connection between the robot nano and robot hc05.
Be sure that the TX and RX lines on both gamepad and robot are hooked up to the correct nano pins.
Also, sometimes you will get a bad jumper wire that doesn't make a good connection. Try swapping in new TX and RX wires.
A way to debug which is having the problem would be to take the robot hc05 and hook it up to a terminal emulator, and see if you get data coming in when you turn on the gamepad. You won't be able to read some of what transfers over because there is a binary protocol happening with length bytes and checksum bytes which are not printable characters. However, you should see things like "W1f" for walking forward or "F1s" in fight mode 1 with no button pressed.
If you see nothing, or you see total binary garbage, then something's wrong with the connections.