Baker was managing director at Maple Leaf Bakery for four years, having previously been chief executive at La Fornaia Bakeries.
Martin Lightbody and David Marshall will step down as non-executive chairman and non-executive director, respectively.
Baker has also held senior chief operating officer and director positions at RHM Consumer Brands, British Bakeries and was managing director of Rank Hovis Mills.
John Duffy, chief executive of Finsbury Food Group, said: “I am very pleased to welcome Peter to the group as chairman. Finsbury Food is at a pivotal stage of its evolution and we have been shaping the expertise and capabilities of the board accordingly. Peter’s wealth of global industry and PLC experience will be a considerable asset to the group.
“I would also like to take the opportunity to thank Martin and David for their support and dedication to the group throughout the years and wish them both the very best for the future.”