Earlier this week Call of Duty: Modern Warfare Remastered's campaign and multiplayer went live, went down then went live once again. Up until that point, the early access version of the game had been made available to select users – not the entire game.
The Call of Duty community on Reddit has noted that servers are available in the North America, United Kingdom, Germany, and Australia. There are conflicting reports, but it appears that downloading and restarting your platform will allow you to play the game.
Of course, to play the game you'll have to have purchased Infinite Warfare and have downloaded it as well.
The full version of Modern Warfare Remastered apparently contains deeper player customization compared to the original game. Modern Warfare Remastered features new emblems, title cards, weapon stickers, and more, as well as 20 prestiges rather than the 10 that were in the original game.