Their appointment is the second major hiring in as many weeks by Nick Bell, the executive creative director at JWT, following last week's signing of the director Tom Carty as a freelance writer.
Six members of the agency's creative department have since been made redundant as part of the revamp.
The move reunites the art director Davis and the copywriter Budds with Bell, their former boss at Burnett.
The pair, who first became a team at Burnett in 1999, were behind the McDonald's spot "my kinda town" and the BTAA-awarded "being six", also for McDonald's.
Budds also won a gold at Cannes for his work on The Big Issue and a Campaign Poster Awards silver for Eurostar.
Bell said: "Jonathan and Anita are a creative director's dream. They are smart, fast and extremely talented. But for me, crucially, it is their attitude - they refuse to let the business grind them down and are always looking for the next opportunity."
"I think JWT really does feel it wants to move into a different gear," Budds said. "Nick now has the chance to do what he did at Leo Burnett and turn an unglamorous agency into something more sexy."