Could be blockage inside the cold pipe. That will need checked first and best way would be to have the float valve on toilet cistern diconnected or taken apart and pipe and valve flushed.
A bit of fibreglass loft insulation, or piece of hairfelt lagging for example, will block the pipe easily and back filling will maybe only temporary unblock the tap. Highly likely.
Another possibility is air drawn into the pipe, but that is easily proved when back flushing because air would be seen and heard coming into cold storage tank in loft.
(I suspect on one job I have been trying to sort, where the gravity cold to bath and toilet has air locked suddenly on several occasions, - I think it was the cold bath tap washer, but that's a rare one.)