Apple has now started uploading full versions of its Beats 1 radio shows on Apple Music's Connect. If you follow a favorite DJ on Connect, such as Zane Lowe or Julie Adenuga, you'll be able to find full replays of their most recent shows, commentary and all.
The new replays work very much like podcasts, allowing you to listen to one "track" that is the show in its entirety. This has its pitfalls, of course, in that you won't have the ability to "love" new songs as they are played like you can do with the live broadcast. Still, this is great for those that aren't able to catch the shows as they are broadcast.
For those who still prefer to catch the tracks featured on each show without the DJ chatter, it looks like you can still find playlists of each show's most recent track list on Connect. If the idea of replays strikes your fancy, however, you'll find them available from the Connect section of each DJ's Apple Music page.