Was she upset because her uncle (had died / has died) recently?
Which tense is correct here, past perfect or present perfect? And also if I changed "was she upset" to "is she upset", does that affect the tense?
Was she upset because her uncle has died recently? Has died : refer to a very recent death.
You could also use "had died" if it happened in the past, or before another action takes place.
For example, Her uncle had passed away when she arrived the hospital.