So today we were playing Battlefield Engineering on the cantina map.

A wookiee who had used the adrenaline card was standing by a component.

Our question was, did the end of round trigger for controlling the component happen before or after the adrenaline wore off?

(If he died before he could controlled the component, I could have swooped in to get it)

Which happens first? This would apply to other scenarios as well I'm sure, such as having a wookiee with 11+ damage by a terminal etc

As far as I know, during a round ends before the status phase, and until the end of round ends before end of round (there's a confirmation about this somewhere). It also matches common sense - you cant have something during a step if it ends before the step.

Even if Adrenaline used "until the end of round", it would expire before end of round effects are resolved.

FAQ:

Q: When an ability's effect lasts "until" a certain timing
window (e.g. "Until the end of the round..."), does that effect
persist through that timing window?

A: No. These abilities' effects end immediately as the specified
timing window begins, before any other effects in that timing
window are triggered. So, in the given example, the effect that
persists "until the end of the round" would expire before any other
"end of round" effects are triggered.