Redlich has been at ABC for ten years, leading a team of 60 auditors. He will build on the changes that Wright made to the organisation as it went through major changes towards a more digital-based offering.
Redlich has previously worked in a finance capacity at Hoilday Autos, and World Challenge: Adventure Expeditions. He qualified as a chartered accountant at Ernst and Young.
ABC is currently looking for a permanent chief executive.
Sally Cartwright, the chairman of ABC, said: "Simon has a wealth of experience at ABC and will provide much needed consistency and continuity as ABC continues to deliver to the evolving needs of the industry."
Wright has led the ABC through one of its most challenging periods in its 84-year history as print circulations for newspapers and magazines have fallen.
The chief executive has attempted to reposition the organisation away from print auditing towards being a digitally-focussed media industry standards and verification provider, operating across a wide range of digital and other channels.
Redlich said: "I’m excited to be taking on this role and look forward to working with the ABC team to further develop the services we offer the industry. We’ll continue to work closely with the ABC Board and committees to meet industry needs and ensure ABC is the media industry’s ‘stamp of trust’.