The handgun purchase permit is not required to posses, or in Omaha, to register a handgun (or other covered firearm). Does it help, yes and no; with it the person behind the desk knows you did a background check as recent as the card is, but if you are not coming from a gun shop with a receipt they will still ask about how you got the firearm. Even if you have the card OPD will run a background check on you because the City of Omaha thinks it can have stricter requirements than the state or Feds.
Next how do you register; if you have only one firearm to register just take it, unloaded and cased, to the main police office Monday through Friday hours are between 8 and 5, outside of those times and I doubt they will have personnel there to do the registration. They do not publicize this on their website but it seems to be the way they work. They can do up to 3 firearms at a time this way, I will discuss this more later. They will require a photo ID with your current address, so get you Nebraska drivers license first.
If you have more than 3 guns to register, you can leave them with OPD to process as they get time or make appointments to bring them in in batches of 3.
Now I mentioned that the city does not allow ownership of concealable firearms by some persons that can via state and federal law, and in fact pass all other background checks. For that reason I recommend only bringing one firearm into the city for the first go around, I have worked with a person where OPD held their firearm, and in fact tried to get it destroyed, while they were appealing and fighting a denial because of a sealed court record in another state. They had done nothing wrong and had been cleared of a bad charge, but there was an issue with police misusing their powers and when everything was settled the courts sealed all the records; because of this sealed record OPD did not want to register this persons firearm.
As mentioned elsewhere in Omaha any firearm under 26" is considered a concealable firearm and needs to be registered, so if you have a SBR etc. those need to be registered as well, even though they are already registered with the Feds.
OPD Q&A on registering firearms:
http://opd.ci.omaha.ne.us/services/citizen-services/handgun-registrationOPD Desk to call to ask any further questions:(402) 444-5818