LONDON — The BBC’s chairman said Sunday that the broadcasting organization was in a ‘‘ghastly mess’’ as a result of its bungled coverage of a decades-old sexual abuse scandal and was in need of a fundamental shake-up.

“Does the BBC need a thorough structural overhaul? Of course it does,’’ the chairman of the BBC Trust, Chris Patten, said on ‘‘The Andrew Marr Show,’’ the BBC’s flagship Sunday morning talk show, after the resignation of the broadcaster’s chief executive.