Moving to NYC as a newly qualified british nurse?

You need to read the International Forum, esp the "sticky" posts. Currently there is no way to immigrate to the US as there are no visas available. There is no way to predict what the situation will be like by the time you graduate. Be aware that you cannot "plan to move" until you qualify for a suitable visa.... I wouldn't worry about apartment costs at this stage.

You will need to ensure that your degree covers sufficient theory and practical experience is the areas required by the US nursing board to which you will apply (peds, med, surg, psych and obgyn/maternity) in order to be eligible to sit the licensing exam.

Moved to the International forum

As mentioned in the previous post, have a good read as there are currently no visas and no idea on how long it will take, currently people have been waiting since 2005 for a visa and still waiting. US has had retrogression since Oct 2006 and demand is still a lot higher than visas allocated

You have a lot to get through and can only be done once you have completed training and have a complete set of transcripts. Be aware that training in the UK doesn't always meet US requirements and I would suggest you ask your university for hours both theory and practical in all areas as it will be expected for the US. Also be aware the university is not obliged to do this as they are training you for the UK not the US

And I will add the following:

NY actually requires that the CVS be completed by CGFNS before they will even look at your papers; this is taking 6 to 8 months to complete and this requires an actual completed set of transcripts.

Planning to work as an RN directly after you finish your training and in NY is going to be impossible. And with the retrogression added in, that we have had in place for more than two years as of now is going to make it even more difficult.