The versatile midfielder/forward joined the Celts on loan from Sheffield United at the end of February.
He has been with the Blades since his early teens, making his debut for their Under 23s side at the age of 15.
Williams originally started off life as a full-back or winger, but has now found himself at home in midfield and as a striker in the U18s and U23s.
The 19-year-old has already gained National League North experience having spent time on loan at Darlington this season, featuring for the Quakers on seven occasions.