Hey Tenno,
Apologies for the late response on this! The PS4 team has been taking a look at the issue of getting stuck on this error message: "An updated version of Warframe is now available. Missions cannot be entered until the update is installed. Please exit and run the launcher."
We have a lead on what may be causing this and are looking to hot-drop a potential fix immediately after tomorrow's ~ 11 AM ET launch of The Deadlock Protocol. We'll be monitoring for any lingering issues once it is live, so that you can get straight to enjoying the update.
Thanks!