Quite honestly, batting out-of-order calls can make my head spin! Just to make matters worse, ASA, NSA and NFHS all have some slightly different interpretations on how this is handled.
I'll have to qualify my answer by saying that my umpire certification is for ASA ball. The NSA rule book is not 100% specific on the point you raise, but the ASA book is.
The ASA rule specifies that if the next batter due up is the improper batter that already batted, and that improper batter was put out as a result of her at-bat, then she DOES NOT bat again. You skip her and go to the next batter. The out recorded against her stands.
If the improper batter had reached base safely, then the BOO was appealed, she would be removed from base and DOES bat again. The result of her first at-bat is ignored. An out is recorded against the proper batter who should have batted.
As for the NSA ruling, my reading of their on-line rule book doesn't totally answer your question. The NSA on-line Case Book does spell out a couple of situations where the improper batter safely reaches base, BOO is appealed and the improper batter bats again.
What I don't see specifically spelled out for NSA is what happens when the improper batter makes an out. The implication would be that, since all outs stand, this would be ruled on the same as under ASA rules.
So, to answer your question: From an ASA standpoint the #2 batter would bat again if she had reached base safely, she would not if she was put out as a result of her at-bat. NSA seems to have the same interpretation, but I would defer that opinion to a registered NSA official.