Reporting to Charlie Rudd, the deputy managing director, Cobbold will lead the global Johnnie Walker account, as well as work on new business.
Cobbold moved to Grey earlier in the year from the now defunct United when it was folded into its WPP sister agency. In the time he was there, Cobbold led the pan-European Toshiba pitch.
He joined HHCL in 2005 as the client services director and had overall responsibility for COI and Bird's Eye.
Ben Fennell, the managing director of BBH, said: "We are always seeking the very best talent, and in Jason we have found just that. His well-rounded skillset lends itself perfectly to our pursuit of creative excellence and ongoing positive client relationships."
Cobbold began his career in 1996 as a graduate at Delaney Fletcher Bozell. After two years at the agency, he moved to BMP to work on the Vodafone business.
In 2001, he moved to TBWA as a partner to lead the 3 account, as well as to work on new business.