This film, part of the BFI Sci-Fi season, also screens on 25 October (with an introduction by John Oliver). Full details here.
Chicago Reader review:
Joseph Losey's black-and-white SF thriller, made in 1962 during his pre-Pinter British period, begins as a sort of love story—MacDonald Carey is an American businessman who shows interest in Shirley Anne Field and as a consequence gets beaten up by teddy boys led by Oliver Reed—then gradually turns into an antinuclear parable about radioactive children sequestered from humanity in an underground cave. The film was mangled by distributors but later restored for TV; more than an interesting curiosity, it's one of Losey's best English efforts, and Viveca Lindfors contributes a striking part as an eccentric sculptress.
Here (and above) is the trailer.