It's hard to extrapolate when you look at the full lore because Metzen is a fucking hack that can't into consistency. Anub'arak was a willing servant in Warcraft 3, akin to Ner'zhul's other generals. He and Arthas spoke on equal terms and were both clearly in control of their faculties beyond the baseline loyalty to the Lich King. So, if that's as far into the lore as you go, he was a loyal servant.
However, WotLK fucks this up by suggesting that he was enslaved to turn on his people. When he's defeated the first time, he expresses relief at being freed from his servitude. The second time he appears, he's basically a mindless husk. Neither are consistent with WC3, either personality wise or lore wise (he's literally the only Scourge to be resurrected from death after being undead).
Personally, I refuse to acknowledge the WoW lore. Metzen ruined dozens of storylines and Anub'arak's was one of the most egregious considering his role in raising Arthas to become the Lich King.