Learn Turkish: Filmi sevmiyor değilim, ama zaten uyukluyorum. - It is not that I don't like the movie, but I am dozing off anyway.

Zaten acts as a discourse marker meaning "anyway" or "as it is." It reinforces the speaker's excuse. While ama uyukluyorum just means "but I am dozing off," adding zaten implies "I was already dozing off regardless of the movie's quality" or "my sleepiness is an unavoidable fact."

What is the nuance of saying sevmiyor değilim instead of just seviyorum?

This is a double negative structure (sevmiyor + değilim), directly translating to "I am not not-liking." Just like in English ("It's not that I don't like it"), it softens the statement. The speaker uses it defensively to ensure the listener knows the movie isn't bad, it's just that they are too tired to watch it.

What is the root of uyukluyorum and how is it different from uyuyorum?

The verb is uyuklamak (to doze off, to nod off), which is derived from uyumak (to sleep). While uyuyorum means "I am sleeping" (a full state of sleep), uyukluyorum describes the action of repeatedly nodding off or struggling to stay awake, which fits perfectly when watching a movie while tired.

Could we put zaten at the very end of the sentence?