Why not start her one morning a week, with you staying for a large chuck of it. Then you can start "popping to the shop" near the nursery and coming back 20 mins later, and so on and so on extending the time you are away until she feels OK with it.
My DD started one morning a week at 2.7 yrs. At 3 she started doing two mornings a week. It gradually increased until she was doing the full 15 hrs in the weeks after the Easter before she started school (she is a June birthday so she was rising 4 then).
She was the kind of kid that took to it without a backward glance too and without a tear, but even so I wouldn't have gone straight in at 15 hrs a week.
My son (2.3) is I think going to be much more clingy and I doubt he will be ready to set off happily to nursery at 2.7. So I will hold off with him if necessary.
If your DD is crying so much she is throwing up, she is pretty upset. It might be that she really is just not ready yet. Of course any new experience will be unsettling whatever age she does it, but crying to the point of vomiting is severe.
In my experience when they are toddlers, i.e. 2 and sometimes 3 year olds too, the predominant theme in their lives is "where is my primary carer, am I going to be abandoned"!! Of course they play independently a bit at home, but that nonetheless is the main thing going on with them. The next stage on is where they have matured sufficiently that "where is my attachment figure??!!" is no longer such a concern. They have language and the conceptual maturity to understand time and space and they know they will be reunited with their main attachment figure and they are not so worried about separation. This frees them up to enter a new phase of really imaginative play, getting lost in what they are doing, often playing in their rooms for ages, and it is this imaginative creative world that comes to dominate, rather than separation anxiety. Its also the time when peer friendships come to the fore, and you may find that if a child can't go to preschool to see her friends she is very upset (whereas when younger the default preference is often to stay at home with you).
All kids make the transition at different times. Some by three are ready to make it (my DD was I think). Others are four or even older; I am guessing my son will be like this. Maybe your DD needs a bit longer (esp. as the new baby can interrupt the process and make them more clingy and anxious about their attachment to you).
Its a bit like potty training - if you can wait till they really are ready its a hell of a lot easier for all concerned.