You mention that you have a stopcock serving only the outside tap.
I would guess that is where the problem is. It's possibly stuck down.
These things work by raising and lowering a valve fitted with a tap washer. Often, when opening the valve, the handle winds up ok, but leaves the washer still stuck on its base.
Turn off the main incoming stopcock. dismantle the outside tap stopcock, and free it up. It's pretty easy to dismantle.