If I understand it right, you have to have a CHP from Nebraska first. So you are out of luck with Utah as Nebraska recognizes it.
What Must Accompany the Application?
•A Photocopy of your Driver License
•Photograph. One recent color photograph of passport quality. *Photos may be taken at the Bureau of Criminal Identification for a $15 fee.
•Non-resident proof of permit. If you reside in a state that recognizes the validity of the Utah CFP or has reciprocity with Utah, you must obtain a CFP or CCW from your home state and submit a copy of it with your application for a Utah permit. For a list of reciprocal states click here. You are considered a resident of whichever state issued your ID. If your state does not recognize the Utah permit this does not apply.
•Fingerprint Card. One fingerprint card. Must be filled out completely. Writing and prints must be legible. Fingerprint should be taken by a trained fingerprint technician. Fingerprint cards that are not legible will be returned to the applicant and will cause a delay in processing the application.
*Fingerprint services are available from BCI or may be offered through your local law enforcement agency. There are private businesses that also offer this service. Check in your local phone directory. Photos and fingerprints may be obtained at the Bureau of Criminal Identification (3888 W 5400 S, Taylorsville, UT.) The cost is $15.00 for 1 photo and $15.00 for up to 3 fingerprint cards.
•Weapon Familiarity Certification. Applicants must complete a firearms familiarity course certified by BCI. No exceptions. The course must be completed before you apply for a permit. Please have your instructor complete the certification information on the application. View a list of (in-state) (out-of-state) certified instructors.