Context: the writer is talking about going to a cinema that has just opened near her house.
朝早く行くと、貸し切り状態なんていう時もあるのでテンションが上がります！ ただ…朝早く観に行く時は確実に徹夜明けなんですけどね（笑）最高の贅沢です♡
Why is going to the cinema early in the morning after staying up all night a luxury? Does 贅沢 have a different nuance here? Thank you for your help!
This 贅沢 means a luxury experience, and it seems to refer to watching the movie in the large theater alone as if she had rent the whole room.
As you suspect, the order of the sentences is a bit unnatural, and 贅沢です could have come right after the first sentence.
The translation of 贅沢 as luxury is not wide off the mark, but it's not accurate either. Sad to say, that's what you get from a Japanese-English dictionary.
If you look it up in a Japanese dictionary, say, the default dictionary provided by Mac OSX (スーパー大辞林), you get different meanings. The first one of them is "必要以上の金や物を使うこと（さま）", things like using money or good that is more than necessary. I think this meaning would fit to the context mentioned above. The second meaning given by the dictionary indeed corresponds to the English luxury, "金・労力などを多くかけていること（さま）". You may need to consider the context to determine which sense is used.
