LONDON — When Tasnime Akunjee’s father, a university student from a middle-class family in Bangladesh, came to Britain in the early 1970s, he thought the streets of London would be paved with gold. Literally.

“In those days, Britain had a powerful story to tell,” said Mr. Akunjee, a London-based lawyer. Once his father arrived, he said, he was shocked to find that he was walking on mere paving stones. But he quickly recovered. Golden sidewalks or not, Mr. Akunjee said, “he wanted to be part of it — they all did in that generation.”

Even educated young men and women in Britain’s former colonies believed in a sometimes absurdly idealized marketing pitch of its former empire. Now, a different pitch, but one that is proving similarly alluring, has swayed hundreds of young British Muslims into believing the Islamic State’s vision of its self-proclaimed caliphate in Syria and Iraq. Leaving behind the Western opportunities their parents came to Britain for, those young Muslims make for a promised land of religious virtue, Muslim community and righteous revolution.