Gareth Edwards had only directed one film before tackling 2014's massive Godzilla. That film got him the gig helming Rogue One: A Star Wars Story, and although he was expected to return for Godzilla 2, that's no longer happening.
Deadline just broke the news of Edwards' departure. They describe the split as "amicable" which seems likely, considering Edwards would have had to move almost directly into Godzilla 2 once Rogue One is finished.
Recently, Warner Bros. and Legendary shuffled around the release dates for their upcoming monster movies, which are slowing creating a shared universe. Of course there was Godzilla in 2014, and Kong: Skull Island will be out in March 2017, followed by Godzilla 2 on March 22, 2019. And then, Godzilla vs. King Kong is slated for a May 20, 2020 release. Max Borenstein wrote the current draft of the Godzilla sequel which is expected to feature the beast battling Mothra and others. They're currently looking for directors.
As for Edwards, we think he'll be ok. Here's a recent picture.