After a lot of bellyaching, it seems as though mods will be coming to the PS4 versions of Fallout 4 and The Elder Scrolls Skyrim: Special Edition after all. However, these mods will be far more limited in their implementation that they are on PC or even Xbox One.
Among the limits that will be imposed on mods on PS4 will be file sizes. For potential modders looking to get the in on the exact specifics of how mods on PlayStation 4 will work for these games, Bethesda have confirmed that they will be sharing specifics sometime soon. So file size limits for PS4 mods of Fallout and Skyrim Remastered will be shared soon, according to what Pete Hines said on his Twitter.
Hines also refused to commit to the implementation of mods on PS4 ever being on par with how it is on PC and Xbox One, noting that the earlier attempt to do that was met with failure, implying that this is the best Bethesda can manage for now. So for now, PS4 players of Fallout 4 and Skyrim Remastered may have to resign themselves to a limited implementation of mods on their platform. I guess something is better than nothing.