Every character that dies off screen "dies eventually". The time line between everyone's deaths and the meet ups at the church are unclear. Also, in the entire purgatory/alternate universe/ideal universe... EVERYONE is dead there.
When they set off the bomb all they accomplished was creating the conditions that led to the hatch in the first place and sent themselves to the right point in time.
No, I believe the alternate universe began when the bomb went off, but everything that happened on the Island was real until you died, THEN you went toward the ALT Universe. The nuke was a reset button, it blasted them back to present time from the 1970s.
The bomb did not trigger the alt time line. The introduction of the alt time line at that point was simply misdirection by the writers. The alt time line was stated as being the afterlife and "timeless", it had no correlation to the bomb other than the editors started telling that story after we witnessed the bomb detonating.
Why did she say 'it worked' before she died, how would she know that?
Juliet thought the plan had worked because she mistook the sideways timeline as an actual rewriting of the past rather than a transitional stage all characters will enter upon their eventual deaths. She really thought their plan to detonate the bomb had worked, and she was understandably mistaken in her belief. Her mistake powered a great deal of the writer's misdirection surrounding the nature of the sideways timeline.