Try what I have suggested, you will notice that the WC will start filling rapidly, wait for it to fill up, then wait some more, keep your hand under the spout the whole time- totally seal the spout off with your hand.
OK, I tried this about three or 4 times, the toilet did indeed start filling quicker, but once I let go the cold tap was still running very slowly. How long should I hold my hand there for?
Let the cistern fill up completely, keep your hand over the spout.....then open the cold basin tap, then a cloakroom cold tap if you have one.