[quote="Simba the Lion King"]So were Simba and Nala injured in their fight?[/quote]
Apparently not. They seem just fine when they recognize each other a few moments later.
As for an in-universe theory, perhaps the dermis of lion skin is thicker than that of a hyena's so a claw scratch wouldn't sink that deep within a lion's skin to erupt the blood vessels within, and explain how Banzai and Shenzi can easily get scratched.
From a production standpoint, it was probably never intended for Simba and Nala to have any scratches or cuts. True, the filmmakers were mindful of giving cuts to the hyenas, but remember in Tarzan when Tarzan's chest was scarred by Sabor and miraculously healed within seconds, the directors of that film stated it was too cumbersome to animate him with those chest scars for a long period of time. I don't know why, but I can probably assume that same situation applies with The Lion King.