Sadly, as of right now the team working on Apex Legends are not working on and have no plans for cross-progression.
I really hope that it is possible for them to implement this system. Thank god they finally announced cross-play, but cross-progression is just as important in a competitive game like Apex where statistics matter.
I mean, let's look at other BRs. I know, I know, Apex are catching up with the latest announcements for fall/autumn 2020, but there is still one huge part missing. Even, even Realm Royale has cross-progression! ☹️