A listing for Call of Duty: Modern Warfare Remastered has appeared on Amazon Japan, which states the game is releasing on June 20 on PlayStation 4 and July 20 on Xbox One. A PC version hasn't been mentioned but that doesn't necessarily mean anything.
This makes good sense. Because earlier GameFly has also listed CoD MW Remastered with the exact same release dates. And we know that Activision has a timed-exclusivity deal with Sony, that ensures every COD DLC comes out on PS4 before anything else. So this could very much mean that CoD MW Remastered will release on PS4 on June 20 and other platforms a month later (i.e. July 20).
Currently the cheapest (legal) way to access the game is buying one of the Legacy editions of Call of Duty: Infinite Warfare, which normally sells for $79.99.