LONDON—When their London Eye gondola suddenly came to a stop nearly 44 stories above the southern bank of the Thames, the visiting Hutchinson family from suburban Boston had no idea why.

“It was scary at first,” recalled Anne Hutchinson, as she described how she, her husband, Jim, and son Jack, from Lincoln, Mass., were stranded high above London for three hours in one of the British capital’s most iconic attractions.

“We...