I was sick just once in labour, when I hit transition. I felt very sick at home before I went into hospital too, but wasn't actually sick (though I had terrible diarrhoea).
I am emetophobic and was really scared of being sick. However, when it came to it, I didn't react to the vomiting at all as I would normally have done. I suppose I had other things on my mind! I just got on with it and, although unpleasant, it didn't make me feel panicky or out of control (as vomiting - my own or someone else's - would normally do) at all.
Incidentally, there was no midwife in the room when it happened. I was in the bath and dh was with me. (He caught it in a cardboard bowl thingy, and a midwife came and dealt with it.)
I think that it would be a shame if the OP's niece was put off being a midwife by this fear, although I understand why she might be (I would be, tbh). Dealing with sick is not a major part of the job, but I think it's fairly clear that it will happen occasionally, and she will need to be prepared to deal with it. I have had counselling for my phobia, which has helped a little bit (at least, with controlling the panic symptoms a bit - not with getting rid of the basic phobia), but I am told that CBT can be effective with this. It's worth exploring, for sure.
Oh, and I agree with whoever it was who said that, if this is emetophobia, it's not the same as just being squeamish. I can deal with any amount of blood, shit, pus, whatever. I just cannot 'do' sick.