Milner, who was unavailable for comment as PRWeek went to press, took up the top PR role last August after acting as head of PR for cash-and-carry division Booker since 1999.
It is understood Milner resigned and is now considering her options, including consultlancy work.
Frozen food firm BFG was renamed last month following the merger of Iceland with both Booker and food service Woodward in May 2000. The firm has now handed Sawday the top PR role covering all external communications.
At Safeway, comms director Kevin Hawkins said he has appointed an interim controller of comms to replace Sawday, hiring the supermarket's former consumer PR manager Janey Berry on a part-time basis.
Sawday's predecessor Alison Bush, who is on maternity leave, may take on the role when she returns later in the summer.
Sawday will report to BFG chief executive Bill Grimsey, who took up the post last year.