As Tjm says, at least part of that story did appear in newspapers last weekend but the story was swiftly pulled from the digital editions.
Even leaving aside the inappropriate angle, the facts that:
* Johnson tried to get his mistress a well-paid job she was completely unqualified for - not the first time he's done this, the only difference being this time he was blocked.
and
* the newspapers didn't just stop the story but actually erased it from digital editions
Both of these being, to my mind, very serious and worthy of follow-up.
As to the veracity of Private Eye. Maybe I'm wrong but I can't imagine the magazine risking a lawsuit on mere gossip. I'd also say that as for their journalism goes, they have a history of covering important stories which nobody else has touched - the Post Office scandal being one.