As far as the Bond movies go, probably not. Churchill died in 1965 at the age of 90; the movie Dr. No was 1962. However, the latest we have record of his reaction to a film is from that year, and it's a pretty interesting one, but I'll save that for the end.

The books? It is extremely likely he read at least one.

...

In April 1917, Valentine Fleming, Conservative member of Parliament, Major of the Queen's Own Oxfordshire Hussars, was stationed on the Western Front in France. He had sent a postcard to his son:

In the wood where we slept last night were wild boars. I killed a snake but not a poisonous one. A hedgehog came into Philip’s shelter one night.

He died a month later at Gillemont Farm (west of Bony; there's aerial photography from that time) from German shells. A eulogy appeared in the Times, ending:

As the war lengthens and intensifies and the extending lists appear, it seems as if one watched at night a well-loved city whose lights, which burn so bright, which burn so true, are extinguished in the distance in the darkness one by one.

The writer in The Times was 'W.S.C.', that is, Winston Spencer Churchill. Churchill and Valentine Fleming were close friends. Ian Fleming, the son who received the postcard, also was appreciative of the eulogy and framed a copy, that he always had hanging.

The Bond author on multiple occasions expressed admiration for Churchill; a 1954 edition of Live and Let Die was sent directly, with handwritten dedication:

From whom I stole some words!

from

The Author

1954

It went up for auction recently, selling for 189,000 GBP, about a quarter of a million in US dollars; the auction site includes some fairly good pictures which indicate the sufficient damage to demonstrate someone probably cracked the spine open. Given how collectors are, it is doubtful the damage came from an owner subsequent to Churchill. So it's likely Churchill read at least one: Live and Let Die.

The "stole some words" part refers to a letter Fleming sent, noting he had "the presumption to steal from Thoughts and Adventures your dramatic tribute to the Secret Service" and that "it makes no demands on the mind of the reader". This was the year after Churchill had won the Nobel Prize in Literature ("for his mastery of historical and biographical description as well as for brilliant oratory in defending exalted human values.") so it is possible he was trying to take extra care to be humble. The jacket flap includes a Churchill quote from the aforementioned book.

As far as what Churchill thought of the book in return, we're not sure; he did write and speak at length of books he liked, including but not limited to 1984, King Solomon's Mines and The Time Machine. (He said of the Wells, "It is one of the books I would like to take with me to Purgatory.") We do know he didn't eschew the middle-brow, and even did reading while WWII was going on, including the first three Hornblower novels. This doesn't quite jump to an outright opinion, alas, or if he caught lines like the one about Bond's housekeeper in From Russia, With Love who would only use "sir" for either an English king or Winston Churchill.

For film, he was allegedly fairly picky, but not in an way where it easy to find patterns. He hated Citizen Kane and it is the only movie we have record of him walking out. His absolute favorite seems to be Lady Hamilton (a 1941 Vivien Leigh / Lawrence Olivier movie with a very pro-British slant). Maybe both can be pigeonholed, but he was also a fan of The Devil and Miss Jones (a department store owner goes undercover to investigate unionizing but romance blooms).

He possesses the ability to shut his mind at will either to worry or work and to live entirely in the present. He can come from a War Cabinet having made a great decision and enjoy an hour of Laurel and Hardy, carrying the weight of the War as if it were a bag of feathers.

As revenge for Churchill's scheduling of (another) showing of Lady Hamilton, the less-appreciative Sir Alexander Cadogan (Under-Secretary in the Foreign Office) scheduled a Donald Duck cartoon and an "appalling slapstick" of Laurel and Hardy (Saps at Sea) and Churchill even realized exactly who did the scheduling.

At the end, Churchill cheerfully declared the pairing "a gay but inconsequent entertainment!"

If he did see Dr. No, perhaps he caught the bronze bust of himself in M's office. The latest we have record of is him watching a film is the 1960 movie Sink the Bismark at the age of 87 (that is, he watched it two years after it released, the year Dr. No came out). The doctor tending him (Dr. Seddon) wrote

I think I watched The Grand Old Warrior as much as the movie. He never took his eyes off it, and they lit up. He sat up and his usually pale face flushed. His cigar went out: he just held it: his mouth opened in rapt attention. Winston was fighting the battle over again.

...

Fleming, F. (Ed.). (2015). The Man with the Golden Typewriter: Ian Fleming's James Bond Letters. Bloomsbury Publishing USA.

Rose, J. (2014). The Literary Churchill: Author, Reader, Actor. United States: Yale University Press.

Wenden, D. J., & Short, K. R. M. (1991). Winston S. Churchill: Film Fan. Historical Journal of Film, Radio and Television, 11(3), 197–214.

The quote from the jacket blurb grabbed from Churchill's book is "In the higher ranges of Secret Service work the actual facts in many cases were in every respect equal to the most fantastic inventions of romance and melodrama. Tangle within tangle, plot and counter-plot, ruse and treachery, cross and double-cross, true agent, false agent, double agent, gold and steel, the bomb, the dagger and the firing party, were interwoven in many a texture so intricate as to be incredible and yet true. The Chief and the High Officers of the Secret Service revelled in these subterranean labyrinths, and pursued their task with cold and silent passion."