Woolf was born in 1921 in Hackney to David and Sarah née Antine and had one younger brother Ralph. In 1939, the family were living at Goswell Road, Finsbury, London and their father David was working as a caretaker to a supply canteen. Woolf enlisted into the RAF at Euston in 1940.

Woolf was the navigator on Halifax DK223 which departed from RAF Holme on Spalding Moor on a night operation to Mannheim and Ludwigshafen. Homeward-bound, the aircraft was shot down by night fighter pilot Hauptmann Heinrich Wohlers of the Stab I./NJG 6, who was flying a Bf 110 G-4 from Mainz-Finthen airfield in Rheinland-Pfalz. Only P/O Arthur Fenton Smith survived. He was captured and became a prisoner of war at Stalag Luft L3.