The new trio will begin filming for the new series during the summer and the show will return later in the year.
Top Gear presenter Paddy said: “I love the fiercely competitive nature of sports stars coupled with having a laugh.
“That’s why A Question Of Sport has always been one of my favourite shows and having been weaned on it from an early age, this is an honour and opportunity I’m certainly not taking lightly.”
Paddy said he grew up watching the show when it was hosted by David Vine and later by David Coleman, but it was Sue who “made this show her own”.
She had fronted the long-running quiz show for 24 years before her final episode was broadcast.
Sam, who won gold at the 2016 Olympics, said joining the show was her “dream job”.
“I’ve watched A Question Of Sport since I was a little girl and my ambition to appear on the show came true five years ago – but I never thought that I would one day be a captain,” she said.
“And the first female captain in the show’s 50-year history too.”
Ugo added: “It’s an incredible honour to be selected as A Question Of Sport’s team captain. It’s a show I have watched my entire life, so to be named as a captain is truly humbling.”
The new-look show will retain A Question Of Sport’s most popular challenges and rounds, the BBC said, but will also introduce some new elements to the format.