BT Home Hub almost certainly goes 1 way - so ADSL in, wifi and cat5 out. I very much doubt stuffing a cat 5 cable inbound will cause the wireless to broadcast. If it does, then their firewall inside the device is even more **** than I expected.
The set up:
Home Hub -> Cable -> Adapter - .... mains wire .... -> Adapter -> cable
will give you a working cable connection in the shed for one device.
If you need wifi in the shed, I would get a cheap AP and plug it into the cable. Give it a different SSID (shed_WIFI), if you turn the wifi DHCP off (to prevent it assigning addresses), then the addresses will be assigned from the home hub. You probably want to keep the AP DHCP on, so that it can get its own address from the home hub.
If you make the WIFI names the same, you will probably have all sorts of hand off issues as the APs know nothing about each other, but have the same name.