The problems are less than you imagine. I'm not familliar with the NY ID card; used to be long guns were a non-issue in NY State (meaning not NY City), so the card, if it exists, is something new. NH handgun law is minimal. The simple way is to find an FFL in UT who will do a basic transfer for you, that is, receive them. Some FFL's want to receive them from an FFL and are a pain. It is completely legal for an FFL to receive firearms directly from an individual. Those who don't likely are concerned whether or not the guns are stolen. I'm sure that is a legitimate concern but, as a practical matter, why bother? Find one who will receive direct.
The FFL will have to send a signed copy for transfer to the individuals shipping the guns. THose individuals shipping can't use the Postal Service for handguns. So, go to UPS. THe law is to ship them overnight, expensive but probably a good idea anyway, even if it were not required. Your local FFL in UT will enter them in the book, make your friend fill out the 4473's and comply with any local requirements. Done.
The gray area is possession, even if temporary, in NY without the ID card. Frankly, and this is not to be considered legal advice or advice condoning what the totalitarians want to call a crime, but I'd just have them sent and forget about the card issue. Once in the FFL book, it's in UT. Note also that there are different (meaning lesser) requirements on the Federal level for firearms over 100 years old. This may also be on the state level, as well.