"It's the worst time to play against them," Williams told BBC Sport Wales.
"Everyone knows that when you lose a manager or get a new manager it brings the squad together and they dig deep and try to impress people."
In Speed's short spell in charge of United, he lost half of his 18 games in charge before Wales came calling.
Speed now becomes Williams' boss at international level and the defender, who has captained Wales in four of their last seven games, is hoping to continue doing so under Speed's leadership.
But his immediate focus is to get Swansea back to winning ways after missing the chance to go second in the Championship a week ago when Millwall held them to a 1-1 draw at the Liberty Stadium.
And Williams is looking to extend Swansea's fine away form which has seen them unbeaten in the last six - including five wins.
"I think it's down to the way the opposition has been playing," he said. "When we play away and they are at home, they try and win the game which suits us a little bit more.
"Us at home, a lot of times they come to not lose which makes for a dull game so we work a bit harder at home trying to do what we do and we want to try and keep the run going away from home."
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The trip to Bramall Lane will see the Swans re-united with former midfielder Leon Britton who left in the summer after almost eight years at the club.
"It will be funny to see him in a different kit," added Williams.
"He's a great lad, but at the end of the day this is business and we'll be going there to get the win and have a laugh afterwards although I'm sure there'll be a bit of banter during the game."
One player who has made the most of midfielder Britton's departure is Joe Allen.
But while the Wales international has stepped up to fill Britton's boots, he will always remain grateful to his ex-team-mate for helping his development.
"We're all looking forward to facing him again," said Allen. "We miss him as a player and a person here and it will be a strange feeling playing against him.
"He was such a key player and the kind of player that played week in, week out and I've admired and learnt a lot from him and I suppose it's paved the way for me to get more game time.
"I've been lucky with the midfielders who have been here since I've come through the Swansea system.
"I've had a lot of players I can learn from and he's probably the stand-out one for me."
Source: BBC Sport
Source: BBC Sport