In 1940, they changed the word Cigarettes to a small font under the red circle. Then, due to the demand for green pigment for the war effort, they changed the color of the pack to white in 1942/43. Lucky Strike was spelled out on the bottom of the 1943 white pack. This was a carry over from the green design.
This kind of package was in use from 1940 till 1943. Text on the tax label says: SEA STORES – For consumption outside the U.S. Penalty for fraudulent relanding or sale in the U.S. So it would be something for the American re-enactor to carry with him in his early days of the war, visiting the UK.