The role was created after the group won a place on the Government’s creative and PR rosters. In August, Engine picked up briefs from HM Revenue & Customs and Royal Mail.
Gallimore will report to Debbie Klein, Engine UK’s chief executive.
At Grey, Gallimore led the British Heart Foundation account, including the "hard and fast" campaign featuring Vinnie Jones.
Klein said: "Engine and its companies have an illustrious history working on government campaigns and we are looking forward to Annie joining to strengthen those relationships."