So what if this plot against Emperor Nero was never discovered and the conspirators under Gaius Calpurnius Piso managed to get rid of Nero? If everything went off right, it seems like Gaius Calpurnius Piso would become Emperor, and the Senate would regain some of the power it had lost in the preceding decades.
Piso seems like he had to be getting up there in years (though his year of birth seems to be unknown?), so there's the matter of who becomes emperor next. He had a son, Calpurnius Piso Galerianus, who like Piso, is a descendent of Crassus. Even more interesting is his cousin, Lucius Calpurnius Piso Frugi Licinianus, who was a descendent of both Crassus and Pompey, but he's rather young (born 38). There's also a variety of other relatives of his and their descendents, some of whom became consuls and other high positions later on in OTL, but Galerianus and Licinianus strike me as the most interesting potential heirs.
Piso himself seems like he wouldn't have made a terrible emperor, but likely not as good as someone like Vespasian. So what are the long-term effects from this? The events of the last few years of Nero's reign will not happen or will happen differently, including the Jewish War of 66.