Taika as revealed in an interview (video below, article here) with Collider representative Perri Nemiroff the location and likely schedule for filming S2:

We start filming hopefully I think by October, and we’re gonna shoot in New Zealand. We shot the last one in LA. Weird place to try and do something all on the ocean, so we’re gonna go down to New Zealand which is surrounded by it. That’s one of the things I’m most excited about for the next year.

https://collider.com/our-flag-means-death-season-2-taikia-waititi-production-start-filming-location-comments/

IndieWire has also released an interview with Taika, who revealed the shooting location was been changed due to budgetary concerns:

On that note, let’s talk about “Our Flag Means Death.” In addition to your starring role as Blackbeard, you’re an executive producer on the show, but even as it had a huge following HBO didn’t renew it right away. Now it’s getting a second season, but the reports are that the budget had to come down. How did you help navigate that outcome?

We just really wanted to make the second season and try our hardest to cut these things down to make it more affordable. Shooting in Los Angeles was so expensive and that’s the reason we’re not going to shoot there again. Until those prices come down for things, they just have to move to other places. So we’re going to New Zealand. I haven’t really looked at the scripts and stuff. They’re still in the story room developing the whole season, but I’m pretty confident that because we’re in New Zealand it’s going to be more affordable. There are more rebates and stuff. It will still have the same scale to it, which you wouldn’t be able to have with a budget cut in Los Angeles. 

https://www.indiewire.com/2022/07/taika-waititi-interview-thor-love-and-thunder-1234738256/

We’re glad that Rhys and Taika (and other cast and crew!) can be home again for S2. As for timing, S1 was shot between June and September of 2021, which is 4 months, so if S2 is on the same schedule, we can hope for a January 2023 wrap date – and then six months of post like S1, for a June release date? We’ll see!