A slightly complicated answer as far as the book goes. Essentially there are two possible interpretations as to the exact year:
Book Answer 1: The first book, The Lion, The Witch, and the Wardrobe, opens with the children being evacuated from London as a result of The Blitz, which began in September 1940. The events of the next book, Prince Caspian, are stated to occur 1 year after the events of LWW and at the start of a new school year, thus PC is set in September 1941. The events of The Voyage of the Dawn Treader are stated to have occured roughly 1 year after the events of PC, but at the end of the school year. Eustace's first Diary entry while aboard the Dawn Treader records the date of August 7th, so we can assume the start of VDT to be early in August 1942. However, towards the start of the book, CS Lewis states in his narration that the events of LWW occured "long ago in the war years". To reconcile this discrepancy with the books own internal chronology, many assume Lewis is making this statement from his own perspective, writing in 1952, that the war years were "long ago" (all the books are portrayed as a narrated story, as if the tale is being told to the reader by someone, presumably CS Lewis, retelling a story he's heard before). Nevertheless, the date of 1942 matches up with the year stated on the official CS Lewis Narnia Timeline, although some doubt whether or not the timeline was actually written by Lewis himself, as it was not published till after his death.
Book Answer 2: The second viewpoint makes the assumption that the statement in VDT about LWW being "long ago in the war years" should be taken as in-universe canon, and that we should thus work backwards from there to calculate the dates of the other books. However, with such an assumption, it is difficult to make the "long ago in the war years" statement fit with the stated time lapses between stories, and with the time frame of real world events, in particular The Blitz, which lasted from September 1940 to May 1941. In order to reconcile this problem, one can take into consideration the idea that the evacuation in LWW was during the Summer, as Lucy told Tumnus this upon their first meeting. There was only one time when the bombs fell in London in the Summer months and that was in 1944. If they were part of the final evacuations, that would push the timeline up four years, and would allow VDT to be set in the post-war years, giving an approximate date of August 1946 (1 year after the end of World War II).
Movie Answer: In the books, the World War II setting is only ever really mentioned once at the start of LWW. Generally thereafter it is largely ignored (save for the aforementioned statement from VDT). The movies however have placed a much greater deal of emphasis on the period setting of the Earth based scenes. For The Voyage of the Dawn Treader they are continuing the World War II period theme with the movie opening on a World War II street scene in Cambridge, UK. At the start of the movie, Eustace writes in his diary that his cousins have been staying with him for over 200 days. Adding on the time they would of spent at boarding school prior to going to stay with Eustace (the start of Prince Caspian shows them heading off to boarding school at the start of a new school year) we can assume the time gap since the events of Prince Caspian is closer to 2 years (rather than the 1 year it is in the book), making it some time in 1943. The detailing on one of the prop Car Tax Discs seen in behind-the-scenes photos of the opening street scene reveal that the year is indeed 1943.