Discounting obvious differences due to the nature of the books and the supernaturals within them, does the core of the combat system work the same? I know it's not recommended for oWOD, but if I were to cross splats (vampire + werewolf in this case) would the systems work roughly the same (even if it is unbalanced)?
Yes, the combat system works exactly the same. The same skills, the same dice pools, all the same.
That said, the power levels are very different. A mage is a mortal and dies like any other mortal. A vampire is strong, but then again no match for a werewolf. Magic is something nobody else is prepared to defend against and the manipulative nature of Vampires is best managed without rules at all. They all have their strong sides, but whoever decides to attack the other in their own game, would probably leave the other without a chance.
Rule-wise you can build a group from any kind of oWod books. They all follow the same rules. But the group will be unbalanced in itself, so you should play a group that wants to group together ingame. Power struggles inside the group would be highly unfair.
I'm not entirely sure why it's "not recommended" to mix the books, we did so for years. If the players want to make it work, it works great. If the players don't, then even a better system will not save them.