Barron, who takes up his role on 1 April, was previously Edelman’s executive director responsible for business development.
Previously, he was managing director of the corporate and financial team between 2010 and 2013. Lundie is understood to be moving to a role at Barclays.
Edelman’s UK and Ireland general manager, Alex Bigg, said: "Talent development is central to our strategy for growth and I am delighted we have been able to find an outstanding internal candidate for this critical appointment. Nick has played a key role in driving the development of Edelman’s corporate and financial practice and he will be leading an incredibly strong team."