It sounds like you have debris in the pipe, which may have got into the tap (if you have already checked the valve), a self cutting valve is not really suitable and should really be avoided.
I would be inclined to cut the pipe using push fit T such as the ones here, then fit a separate stopcock, before taking the pipe to the outside tap.
Should you wish to proceed as is, take the tap off the tap connector, turn on the under sink valve and see if water will gush out of the tap connector for a good minute, if it does switch it off, then strip down the tap, if it doesnt, check all the other taps which are up stream of the newly fitted valve work.
Let me know how you get on.
Hi Oran, I just wanted to follow up and see if I can be of any further assistance.
Hi Oran, so I can close the case please login and rate the service received using the star rating on the Just Answer page, should you need further assistance please let me know.