I'm not sure when Apple rolled out this change, exactly (though it appears to have gone live over the past three days), but you can now listen to several Beats 1 shows (including those by Zane, Julie, and Ebro) in full through Beats 1 Replays.
Previously, unless you listened to a show live or during its rerun twelve hours later, you were out of luck. Some of the shows on Beats 1 offered the ability to follow them on Connect and listen to the playlists of songs played during the show, but that didn't include the complete show with DJ and guests.
Now, if you've missed a show and would prefer to listen to it whenever you'd like, you can find the Replay (again, through Connect) and tap to start listening. The easiest way to find Replays is to look for the DJ's curator page on Connect (such as "Julie Adenuga on Beats 1"), tap the Connect tab, and look for a playable Replay. This way, you'll be able to listen to a show in its entirety without having to skip the DJ talk and interviews with the playlists – which are still available if you only want to get the songs from each show.
Unfortunately, listening to Beats 1 Replays won't display information about the currently playing song as it's the case with a live show on Beats 1 (you can try this by listening to Beats 1 live and checking out song titles in Control Center).1 This means you'll lose the ability to "love" songs while listening to a replay and, of course, you'll need an additional app to identify song titles as they're playing (I recommend Musixmatch and Shazam).
Overall, this is a first step to make sure there's a better interaction between listeners and shows on Beats 1. Besides song metadata while listening to replays, I'd still like to see a full schedule (with clear time zone details) in the app, and it'd also be nice to receive push notifications for when a Beats 1 show from someone I follow on Connect is about to go live. It's good to see that Apple is constantly tweaking Beats 1 and experimenting with Connect, and I hope to continue to see more of this.
Basically, Beats 1 Replays right now are like podcast episodes without chapter marks. ↩