Cope joined Wagamama in October 2013 from pub business Mitchells & Butlers, who he had spent nine years as marketing director. Earlier in his career, he worked as a brand manager for Britvic.
He will sit on the executive board at Byron and report to acting executive chairman, Dalton Philips, who is best known as the former chief executive of Morrisons.
Byron is currently searching for a chief executive, after Andrew Manders stepped down earlier this month, citing personal reasons. Manders had succeeded Byron’s founder Tom Byng, who himself stepped down in December, continuing his involvement as a shareholder and adviser.
Cope said: "As the pioneers of proper hamburgers in the UK, Byron is uniquely well placed to take advantage of the significant growth opportunity that exists for premium brands in the UK’s casual dining market."
Both brands are owned by private equity firm Hutton Collins Partners.
Philips added: "Coming from within the Hutton Collins family, Simon brings with him a wealth of experience and great ideas, with a first class track record of delivery."