Scotland winger Burke, 23, is the Blades' fourth signing of the week and has agreed a three-year deal.
Republic of Ireland international Robinson, 25, has signed a five-year contract with the Baggies.
He spent the second half of 2019-20 on loan at West Brom, helping them achieve promotion to the Premier League.
"I can't wait to get going," said Robinson, who scored three goals in 16 appearances during his stint at The Hawthorns.
"Discussions were taking place over the summer but these things take time. I'm thankful it has all been completed."
"We needed a touch of pace and Oli certainly gives us that," said Blades boss Chris Wilder. "We've been working on it for a few weeks and obviously we wish Callum all the best. It's a move that suits all parties.
"We are delighted to have Oli in the building. He's had a couple of big moves in the past that haven't gone as he'd have hoped, but it's down to us now to give him a kick-start."