It's expected to pass with cross-party support.
Cabinet was still considering the buy-back scheme, which is estimated to cost the Government up to $200 million.
The amnesty to the end of September means that people who own gun parts that will be illegal can hand them to police without fear of prosecution.
Nash said MSSAs or assault rifles were not needed on farms or for hunting, but were made to kill people. He said they had no place in New Zealand, and he had no issue with the speed with which the legislation is being put through.
A second tranche of changes to gun laws would be brought in later and go through the proper legislative process, he said.
It is expected to be introduced later this year.
It will consider whether a national register of firearms is needed, the vetting process for a "fit and proper" person to obtain a gun licence, and the police inspection and monitoring regime, including rules around storage.
Peters said the cost of the buy-back scheme would be outside of existing baselines.
Around 200 firearms have already been handed over to police so far, and about 1400 calls have been made to the police 0800 311311 line.
Nash said anyone can hand in any weapon to police if they wanted to, though the buy-back would only apply to weapons that will be prohibited.
The bill will be sent to the Finance and Expenditure select committee, which has members from every political party, to hear public submissions.
Labour deputy leader Kelvin Davis had handed in a .22. Peters said he did not know if any NZ First MPs had handed in any firearms.
"The misuse of semi-automatic weapons has caused death and injury at our places of worship," Nash said, referring to the Christchurch shootings.
"The men, women and children who died and suffered injuries at the mosques now have their own legacy. We will tighten gun laws to improve the safety and security of all New Zealanders. Their memory is our responsibility."
Peters, who is Acting Prime Minister while Jacinda Ardern is in China, has previously been resistant to tighter gun laws, but has said that the terrorist attack on Christchurch last month had changed everything.
The Government plans to ban on military-style semi-automatic (MSSA) weapons and assault rifles, though there would be exemptions for guns commonly used by farmers for pest control, as well as hunters.
The exemptions would be semi-automatic .22 rifles (with a magazine which holds no more than 10 rounds), as well as semi-automatic or pump action shotguns with internal magazines (holding no more than five rounds).
It is understood urgency will not be needed in the House tomorrow because all parties will support a motion for the bill's first reading. Urgency will be needed next week to pass the bill into law on April 11.
The bill has support across the political spectrum, though Act leader David Seymour has voiced objections to the use of urgency and is unsure if he will vote for the bill yet.
The Government is also working on the details of a buy-back scheme, estimated to cost up to $200 million.
Meanwhile a petition to Parliament urging the Government to take more time to consider the reforms reached 13,700 signatures this afternoon.
Last week, Gun City emailed its subscribers and urged them to sign the petition, which is in the name of Hayden Livingstone.
It called for a longer public consultation period "to ensure effective firearms legislation reform that does not unduly punish law-abiding firearms owners while maintaining assurances of public safety".
It called the reforms "ill-advised, partly due to the speed at which they have been implemented and also due to (understandable) emotionally driven public pressure".
The Gun City email told gun owners to write to MPs, police, and media, and included a link to a Council of Licenced Firearms Owners document to help people prepare submissions.
It also asked them to demand information on the Government's proposed buy-back scheme, including whether it would compensate for gun accessories and ammunition, lost income or employment, booked travel to events to use MSSAs, or losses from investment in shooting facilities.
"The reasons and activities for which we own guns vary. Let's stand united as Firearms Licence holders rather than dividing into different groups, in the hope that other shooting disciplines will not be affected," the email said.
Gun City boss David Tipple has previously confirmed that Gun City had sold four weapons to the accused gunman in a police-verified, online process.
There are 13,500 known MSSAs in New Zealand, and an estimated 1.2 million to 1.5 million firearms overall.
