Ah, all is now clear. IMJ and I were under the impression you had something like these http://www.panasonic.ae/EN/Manuals/KX-TG6821.pdf
So, with the faceplate of the master removed, plug the corded phone in the extension socket. There should be no dial tone.
@Longshanks your link explains how phone is conencted
it helps in we have clarified how your phone connects and any noise which friends were experiencing does not appear to be because your extension is wired incorrectly
really need to stay conencted to test socket with a filter and see if your friends still complain about noise on your phone line if they do then you know that it is not an internal problem so very unlikely to be charged £129.99 for the engineer callout
@Longshanks Yes, it does.
Complete sequence to follow to ensure problem is external to your house.
Remove faceplate and connect corded phone to test socket, no filter required. Check for noise on line. If clear, fault is in your internal wiring. If noisy, remove corded phone from test socket leaving faceplate removed. Connect corded phone to extension socket and ensure there is no dial tone. This will prove that your internal wiring is correct and no longer connected.
Report noisy line fault to BT.
Your upstairs cordless worked because the base unit with the corded handset was connected to the test socket giving the dial tone. Be interesting if still in test socket if incoming calls still get noise problems
Ignore the cordless phone, it is a total red herring, it is not connected directly, it connects to the line via your corded phone.