Not sure about Hudson Mk.III, but probably it entered into service in 1941, as No. 269 Squadron chronology seems to show.
Hudson-MkIII Hudson Mk.III, 1941
"The Hudson Mk I began squadron service with the RAF Coastal Command's No. 224 Squadron in the Summer of 1939. By
September, No. 233 Squadron was similarly equipped,
while No. 220 Squadron had begun to replace its Avro Ansons with the Hudson Mk III. Not long after war broke out, Hudsons also equipped No. 206 and 269 Squadrons. These squadrons all flew maritime patrol and anti-shipping sorties from the British Isles.
Additional squadrons were formed during the war until the RAF saw a peak of 17 Hudson squadrons. Other Hudsons flew reconnaissance missions over Germany, occupied Europe, and (in civil registration) southern parts of the Soviet Union."
"...a Hudson Mk III of No. 220 Squadron guided a British destroyer to the prison-ship Altmark..." ( February-march 1940 ).
"...Mk V... . After the Lend-Lease act of March 1941, aircraft were delivered under USAAF designations."
http://uboat.net/allies/aircraft/hudson.htm various other sources: ".. squadron 220's conversion to Hudsons was completed by November 1939".