Publishing was, as you say, heavily restricted by paper shortages during the war (and millions of books and magazines were destroyed by German bombing), but it was not entirely stopped; new S-F books by both American and British authors (e.g. Edgar Rice Burroughs and Olaf Stapledon) continued to appear in the UK, and the British magazine Tales of Wonder that had started in 1937 continued to be published until 1942 (albeit with an ever-shrinking page count). Pre-war, American and British magazines were available from the Science-Fiction Service, both current and back issues (their 1938 catalogue is available online), though again the war disrupted services as staff were called up.

Probably your best chance of getting your sciencefiction fix would have been through fandom. There was a small but burgeoning UK fan scene in the 1930s with a number of societies and associations, several producing fanzines (or fanmags, as they were generally termed at the time). Most were dissolved on the outbreak of war, but fan activity continued, even publication of fanzines, most notably Mike Rosenblum's Futurian War Digest. Through its pages were tales of misfortune, such as Sidney Birchby's home being destroyed in an air raid: "Imagine my horror! The land's premier collection of Weird Tales scattered over the entire neighbourhood! A Brundage cover in every back garden! [...] Three days, of frantic grubbing under the ruins led to the salvage of about 30% of the collection. Much of it was the worst 30%; stuff by the cheapjacks of s-f; while "lo!" and the best Astoundings went to feed the earthworms." (Birchby's mother was also killed in the blast, of which he made no mention, though that's less callousness than the 'stiff upper lip' attitude of the time). There was other fan news, reviews, information on new publications in both the UK and America where available, sometimes other fanzines included with the distribution, opportunities for fans to get in contact with each other, and to exchange or borrow reading material. This could result in impressive, though very localised, libraries; Frank Parker, working with the fire service, found like-minded folk with similar reading habits during off-duty hours resulting in the Paint Research Station Science Fiction Library in Teddington, expanding to become a wider club.

Contact between British and American fans had started before the war, with letters from British fans appearing in magazines such as Wonder Stories, and continued during the conflict, being one source for both new magazines, and paper for fanzines. American fan John Cunningham founded a "British Science Fiction War Relief Society" in 1941 to supply free pro magazines to S-F starved Brits, and Alden H. Norton, editor of Astonishing Stories and Super Science Stories, also distributed free copies to UK fans.

Rob Hansen's THEN - Science Fiction Fandom in the UK: 1930-1980 is an excellent history of the time, and his site has a phenomenal archive including the Futurian War Digest and other wartime fanzines, and a volume specifically on wartime fandom: HOMEFRONT - Fandom in the UK: 1939-1945