How do I claim the No Time To Explain pulse rifle?
Solution 1:
Now that Beyond Light is released, Time to Explain is now obtainable. In order to receive it, you simply must finish the campaign and then speak to the Exo Stranger.
Looking at the Destiny 2: Beyond Light Deluxe Edition page on the PlayStation Store, the pulse rifle No Time to Explain, along with the other items you mentioned, are not released yet.
The emblems, emotes, and shells you received were apart of the pre-order bonus and were obtained immediately, but the other items not will be accessible until the expansion is released.