After fears the free upgrade would be limited to the PlayStation Plus Extra/Premium version, and Bethesda Softworks removing mention of the upgrade being free for all, it turns out PlayStation Plus Essential owners can get the upgrade for free.
As explained by reddit user Dubsified, PlayStation Plus Essential owners can receive the PlayStation 5 version of Fallout 4 for free by doing the following.
Start by finding Fallout 4 in your Game Library, followed by selecting it. This should take you to the product page. On this page select the … icon and select view product.
On this page there will be a drop down called Game, with another option also called Game. When doing the second option it will let you add Fallout 4 on PlayStation 5 to your library.
We can personally confirm this method works, so anyone waiting for it should do it as soon as possible. There also doesn’t seem to be an official announcement, but assuming this isn’t an unexpected glitch, expect normal methods to work in the near future.
Update: Bethesda confirmed the fix is intended, and now live.
Worked great for me, thanks.