Miller will stop working for the firm on 31 March, while the craft brewer's current owner SABMiller has confirmed the brewery will continue to be run independently by the new GM.
Edwards was at the forefront of Miller Brand's premium beer portfolio and boosted the company's reach in the category as director of brand marketing.
Nick Miller was the managing director of Miller Brands, the UK arm of SABMiller, until 2011, when he left to join Meantime. SABMiller then purchased Meantime in May 2015, before the company changed hands again following the deal with AB InBev last November.
Numerous brands, including Peroni, Grolsch and Meantime were then set to be sold off by AB InBev, in order assist in meeting competition requirements to ensure the merger was completed. As a result, Meantime are set for ownership by Japanese brewery Asahi following the finalisation of the proposed acquisition of SABMiller by AB InBev.
Of his departure, Miller said: "It has been a privilege to lead such a wonderful group of people at Meantime and to work with a great group of other customers, as we have taken Meantime beers to more and more outlets and built a loyal following both here in the UK and abroad."
Edwards noted Meantime's growth since launching in 2000 and said Miller has taken beer production to new heights.
"I have long been a huge fan of the brand so this appointment is a huge honour for me and I am looking forward to working with the team in Greenwich to continue building on the success of Meantime in the UK and beyond."
Meantime was acquired by SABMiller for an undisclosed sum last year. Edwards' enthusiasm to grow the brand's global presence fits with plans announced by the beer giant at the time of the acquisition to look to export the craft brewer's products.
Building on success
Sue Clark, managing director of SABMiller Europe, said: "I am looking forward to working with Laura to continue the strategy of building on the success of Meantime and we are delighted to be able to appoint a fantastic candidate from our internal talent pool with a great passion for beer and outstanding commercial capabilities.
"Since SABMiller purchased Meantime Brewing Company last year, we have ensured that it has continued to operate as a stand-alone business within our wider organisation.
"This will continue to be the case under Laura's leadership. Meantime is at the vanguard of innovation of both beer styles and flavours and is well placed to continue to take the beer category to new consumers on new occasions."
Meantime revealed its seasonal beer line-up earlier this week.