alertly; vigilantly; without letting one's guard down
Adverbial form of 油断 (carelessness, negligence) with なく, meaning 'without being careless'. Used to describe acting with constant attention and caution.
He was vigilantly watching his surroundings.
It is important to proceed with preparations without letting your guard down.
警戒して means 'being on guard' or 'being vigilant', often against a specific threat. 油断なく is more about a general state of not being careless.
Standard kanji spelling.
Alternate kanji spelling using 無く instead of なく.
The adverbial expression consists of 油断 “carelessness; negligence” and なく, the adverbial form of the negative adjective ない: literally, “without negligence.”