Ooo, linguistic tiems.
Given that Primarchs seemed to learn languages fairly quickly and in a way that resembles first language aquisition, I think I can assume they wouldn't have a problem with accents as well. So, if they would use one, it would have some other motive.
For example, I can imagine Horus incorporating a (slighter) accent when giving a speech to a particular group to show.
On the other hand, I figure that Fulgrim would aim for having the equivalent of an recieved pronounciation in High Gothic.
The Primarchs that had a grudge with the Emperor/close ties to their planet might maintain their accent as a sort of demonstration.
So: Guilliman, Dorn, Horus, Sanguinius, Magnus, Lorgar, Perturabo and Fulgrim would be fairly likely to have the equivalent of RP in High Gothic, with Horus occassionally converging to the populace of a planet.
Mortarion, Angron, Konrad Curze, Jagathai Khan and Leman Russ would probably speak with a noticably different accent, which might get more pronounced when speaking with the Emperor in Mortarion's, Konrad's or Angron's cases.
Alpharius and Lion would probably be switching accents depending on what they think would benefit them most ATM.