Of course we don't know; they didn't think enough about it to write it. I base the "Not much" on how the Material Archive artbook has an entry attributing Vergil's quest for power to his desire to protect his mother and what he valued about Sparda's legacy, and then DMC5 bastardized the quest and two halves of Vergil's character down to nothing but a toxic sibling rivalry because Dante had everything Vergil thought he wanted including his mother's love and protection, and it was given validity with outside-POV blurbs about how Vergil would frequently leave the house by himself to read, the Visions of V manga where Eva apparently hasn't taught Dante the basic meaning of boundaries and coddles him, and in the game Dante recounts the flashback of Eva's death in which Eva apparently can't tell when Vergil is in the house or not, even though the manga shows she should know since he was attacked shortly after he left the house in a huff because he couldn't read his book and she let him abandon the book and "self-soothe" instead of running after him. She somehow thought one eight-year-old child as more self-reliant than the other. So yeah, they can say that Vergil loved Nero's mother, but if it's going to be treated in the same way as Vergil's love for Eva (and hers for him), then no, no he doesn't.We don't know that, though. We don't know about the events that led Vergil this way nor why he left whoever she was. In the original Japanese version Vergil doesn't say 'I don't care about my son.' He basically says 'Son? I don't know of any son.'
You know the reason Vergil left to cut the tree down was for Nero, right? Even if he was able to keep fighting and just decided to stop and go was for him. Even if he fought him beforehand Vergil's actions at the end have to do with Nero.
He said something different in Japanese. Okay, and? I still take issue with the idea that he wouldn't even know he had a son, given all the steps that would involve for what's "meant to be" commitment/love. So he'd just have sex with a woman and not know it left her pregnant? He'd pass off his union with a woman as something that "happened a long time ago"? They only needed to change a line or two to express a different idea with Vergil's mindset than what they did.
"Vergil cut the tree down for Nero"? Vergil cut the tree down because he already reaped the benefits of having the tree around, gained more by Urizen reuniting with V, then got his ass kicked twice in a row. Instead of immediately fixing the Qliphoth as soon as he was reformed (or while he told Dante to "heal his wounds and become stronger") he kinda just stood around like a bonehead patiently waiting for Dante to come and fight him to the death, and only pivoted to Nero within a minute or five of Nero showing up to body him. If he cared about Nero's existence or even the tree interfering with the fight, he wouldn't have continued the fight with Dante while the Qliphoth remained standing that then prompted Nero to break it up, and then pick a fight with his kid, he'd have just cut the tree. You're free to read into the drivel the games provided complete with shoddy dialogue and rapid "developments" in character where others don't matter all in the name of subtlety but that's not what I'm seeing with the material that's actually there.