There was
https://en.wikipedia.org/wiki/Northern_Songs
and then there was
https://en.wikipedia.org/wiki/Apple_Records
The former company "owns" the older songs and the later the later songs (that is, the "publishing rights"). As far as I understand only the former company and songs (but that was a significant part of all Beatles songs) ended up by Jackson. The article from the Daily Mail you link to is wrong (now who would expect that) saying "Payday: Sir Paul will be able to receive royalties from Sgt Pepper's Lonely Hearts Club Band and Abbey Road albums" because as far as I understand Abbey Road is an Apple Records album, never owned either by Northern Songs or later Jackson.
Now how I know that, which were the Apple Records records? I'm just old enough to have had the vinyl records, and holding them it was immediately obvious which is which:
http://thebeatles-collection.com/wordpress/wp-content/upload...
vs
https://en.wikipedia.org/wiki/Hey_Jude#/media/File:Hey_Jude_...
Paul apparently has to pay to Northern Songs when he plays his older songs, according to his own words, but is it true?
I guess the "LP vs CD rights" story you mention comes from the fact that the newer remixes were made specially for the CD's. But who owns what when remixing the song "owned" by Northern Songs, I don't know. Anybody knows?
I thought that the Beatles' story is ultimately about how clever their rights management was in the end -- supposedly they only sold the rights for records and tapes, and then when CDs became a thing they got to sell them again. Most recently and publicly they reached a deal with Apple for distribution via iTunes. Exactly what was Apple negotiating for if they had lost ownership of their catalog?
According to Snopes, they sold the publishing rights to Northern Records.
"Paul McCartney and John Lennon (and Lennon's estate, now that he's dead) have always received their 50% songwriter's share of the royalties for all Lennon-McCartney songs."
http://www.snopes.com/music/artists/jackson.asp
Basically, the Beatles tried to be both the artist and the publisher (Northern Records) and after signing over their publishing rights to Northern Records, lost control of Northern Records. So they're reduced to the position of a merely very successful artist who doesn't own his/her own record company.
I also see here that rights revert to the artist after 56 years:
http://www.dailymail.co.uk/tvshowbiz/article-2394325/Sir-Pau...