My DC have been on Nutramigen for a good few years.
DD from 8 mo and I had to introduce it gradually byt swapping a spoonful of cow and gate for nutramigen every day until she was on it fully. If I remember rightly I think it took her about a week to start feeling better. But DD did have alot of colic for the first few months, so as mmmmsleep has suggested, try and get a referral asap but I'd be asking to see a pediatric allergist. They usually have specialist dieticians in their clinics and they are much better than just the dietician at the local hospital, unless the hospital is one of those that deal with allergies.
DD has alot of food allergies and I think her colic was maybe an indicator for this so I'd suggest getting a referral as soon as possible. As your GP has given you nutramigen he/she obviously thinks there's something going on, so strike while the iron's hot, so to speak. Your GP has to refer you if you ask.
If your DDs poo has firmed up after just one day, that could be a good sign as it means it's not going through her too quickly. If she is straining when she goes then obviously it could be constipation which isn't so good.
Does she cry all the time or at certain times of the day? DD's colic cry would start at about 6.30 and would last until about 9.30. If it is a colic cry then some infacol a fe minutes before a feed can help, but I'd speak to the GP about this too just to make sure there isn't anything else it could be.
I hope you're ok. A crying baby can be exhausting so I hope you're getting some rest and making sure you look after yourself and eat well.
take care