I don't expect this will be a problem for you, due to the way DE implemented Cross-Play prior to this new Cross-Save system.
Since the release of Cross-Play, you'll notice that when you view your in-game profile in Warframe, you'll see that appended to the end of your account name/alias is a # followed by a 3-digit number, which is a unique identifier that differentiates Warframe accounts with that have the same name.
So, now when you log onto your Warframe accounts on the different platforms, you'll notice that even though the account name/alias may be the same, the 3-digit number after the # symbol will be different.