If Professor Xavier can't read Magneto's mind because of the helmet of Adamantium, then how he can read the mind of a Wolverine's Adamantium skull?
His helmet is not made of Adamantium.
As mentioned in the answer at another SE site,
In the comic canon, Magneto's helmet is made out of whatever metals Magneto has available. Any strong metal appears to be usable.
He reshapes the metal at the molecular level so it has the anti-psionic abilities necessary to keep his mind from being or affected by telepathic abilities. The metal is likely highly magnetized, since strong magnetic fields reduce or prevent telepaths from being able to read minds.
It is also mentioned here.
Magneto may be able to create his signature helmet by manipulating any metal into the shape, this can be assumed as despite the helmet being destroyed or taken from his possession Magneto always has another.
As mentioned here, his helmet is not made of Adamantium and is designed to stop telepaths from getting into his mind. Therefore, Professor Xavier can't read Magneto's mind but can read the mind of Wolverine's Adamantium skull.
In the X-Men movies, the helmet has two origins. In the original trilogy, it's active technological defense against Xavier, using similar ideas behind Cerebro. Xavier and Magneto built Cerebro together, and after their rift, Magneto built the helmet. Xavier says as much in the first and third movies.
In the First Class movies, the helmet is taken from Sebastian Shaw, who was given it by the Soviets for unknown reasons, obviously to fight against telepathic attacks of some form (Xavier isn't the only telepath in the universe). In these movies, Beast built Cerebro alone. Magneto had departed before that. As we see in First Class.
At no point is it stated or implied to be Adamantium.
In the source material, Xavier has always been able to talk to Wolverine, regardless of the Adamantium in his skull. Unlike another character, Juggernaut, who has a non - Adamantium helmet that does stop him.
Additionally, if we consider that Adamantium like Wolverine was created in an attempt to recreate Captain America's shield, in the 80s or so, then Magneto helmet really can't be Adamantium. The unique metal has not yet been created.