The facts contradict themselves. One set of developers say that 4 is 10 years after 3, and the scenario director says Nero is Vergil's kid. The localization team agrees with the latter, and the press agrees with the former
The game itself hints that Nero is Vergil's kid, from saying he has the "blood of Sparda." to him echoing Vergil's lines ("Scum!" "Give me power!") Note that they say the blood of Sparda, not the power of Sparda. Then there was Dante's
insistence on keeping it in the family, and then letting Nero keep Yamato anyway. But the timeline says it can't be, resulting in a massive continuity snarl.
I think this is a major reason for the reboot, to get a clear and concise timeline.
To me, Dante saying Yamato has to stay in the family (which is why he won't give it to Nero at first) tells me Nero might not even be related to Dante and Vergil. Only when Nero says ''I need this'' does he give Yamato to him. Of course, Dante could mean that he simply views Nero as not being family, since he might be an illegitimate child of Sparda. That doesn't seem likely, though, since Sparda died before Dante and Vergil were even born.
Nero using Vergil's phrases doesn't mean that much, really. He may just be tapping into Yamato's 'soul', as it were. That, or Capcom just wanted to show Vergil is still there in memory and decided to make him say some of Vergil's lines as a wink to the fans. People keep saying Nero's DT looks like Vergil. I say it only looks like him in color, since it's blue. The rest of the DT is derived from Nero's artwork as you can see when looking at my avatar.
The blood of Sparda, well, that could mean anything. It might be meant literally somehow, or he is indeed related to Sparda in some way. I agree, though, that the whole thing may have been poorly thought out by Capcom, and that Nero could be Vergil's son.
It just seems unlikely given Vergil's stance towards humans and him wanting nothing but demonic power, relinquishing his humanity in the process, not to mention the timeline not fitting. Furthermore, why would Vergil's son wear red and blue instead of just blue or a different color?