First stop has to be the Ealing comedies. There are some absolute classics in there but it also captures the feeling of that post-War period.

The Spy Who Came in From the Cold is also well worth a watch, probably my favourite spy movie. Just don't read too much on the plot and

If you want less serious espionage then Carry on Spying is for you. A number of the early Carry Ons are in B&W, although they tend not to be as good as the colour ones where they really hit their stride but some, like Spying, have a lot of charm and Carry on Nurse feels lime them finally honing thr formula. If you like that kind of comedy then try the St Trinian's films.

The Third Man is a classic. You'll also find a number of early Hitchcock films that meet your criteria (The 39 Steps, The Lady Vanishes), as would early Powell & Pressburger (eg Black Narcissus) and.David Lean (Brief Encounter, Great Expectations, Oliver Twist, etc).

The Railway Children is one of those quintessentially British films. As are Went the Day Well? and Billy Liar, although for different reasons.

If you are looking for horror then try the anthology film Dead of Night, that last segment will stay with you. If you want more scares then the early Hammer Films are B&W. Although they are famous for their monster movies don't overlook The Quatermass Experiment or Hound of the Baskervilles. If you like a but if Holmes, then check out the Basil Rathbone ones.

There are countless B&W war movies but I suppose thr one to start with is The Dambusters or Ice Cold in Alex or The Cruel Sea.

Quick.mixed bag of real life or adaptations: A Night to Remember (the best Titanic film, probably), Lord of the Flies, Village of the Damned (love John Wyndham), Things to Come (based on the HG Wells novel), The Scarlett Pimpernel (and let's not forget "Pimpernel" Smith), The Elephant Man by David Lynch, Our Man in Havana (more Graham Greene), and Olivier's Hamlet

If you are up for more modern films, then I'd recommend A Field in England and Control (about Ian Curtis from Joy Division).