Congratulations. life is going to be quite exciting for you and yours next spring and summer.
OK, here's the definitive advice - Delta 3200 rated at 1600, developed in DDX, at the time recommended for 3200 rated film. A nice fast lens, on your 35mm camera. You might want to pack your MF and bring it along, but the ease of use of a 35mm trumps the big negatives. You will undoubtedly be excited and emotional. The MF can be used once things settle, or once the family gets home. You will be dealing with dodging health care workers, trying to catch definitive moments, and varying light levels - it is quite bright on the infant warmer, the scale is usually in the spill of light from the warmer, but the bed where mom will be cuddling babies will be less well lit. As you don't know if the birth will happen at 2 in the afternoon or 2 in the morning, it is hard to predict the room light. There is always the possibility with twins that one or both babies might need to spend time in the nursery, and you will be taking the first photos of them in there.
Flash will not cause permanent damage to the babys' eyes, but is a rude and nasty thing to do to a newborn.
Why is this the definitive advice? Because I'm a midwife.
Have fun!