A suicide bomber who killed 22 people at a Manchester pop concert likely had the help of a terror network, U.K. authorities said, and his brother confessed to a Libyan militia that the two of them belonged to Islamic State.

The allegations came as a portrait emerged of how Salman Abedi, the 22-year-old perpetrator of Britain’s deadliest attack since 2005, grew up straddling middle-class Britain and the tumult of Libya, playing street soccer as a schoolboy before heading off as a teenager to fight alongside his father in their...