From August 21st after doing the cross save, will I be able to play immediately on Battle.net or will I have to wait for October on Steam?
Also if I do the reverse (from PC to PS4) can I play immediately on PS4?
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As long as you own licenses for the content you are playing on the platforms you want to play on, you can play immediately. I suggest you read up on it a bit https://www.bungie.net/7/en/crosssave