National's local government spokesman Chris Luxon said the change was "hidden quietly in the Covid Response (Management Measures) Legislation Bill".
"The bill allowed the Government to adjourn local elections multiple times without constraint – and not just in an election year, but the year after as well," he said.
"The apparent reason for this draconian change was so local elections can be more flexibly managed during a pandemic – forgetting that both the Government and councils already have flexibility around elections," he said.
The Government responded to those criticisms and removed the local government election provisions from the bill. A Government source confirmed the idea of a measure to delay those elections further is now off the table.
"The Government has now buckled under the pressure and will delete that clause from the bill," Luxon said.
National's Covid-19 spokesman Chris Bishop said the proposed change was "ridiculous".
"We argued it was outrageous Labour would use a sweeping Covid bill, being rammed through Parliament under a shortened timeframe, with almost no proper public consultation, to make an enduring electoral change," Bishop said.
The decision comes as the Government faces extraordinary pressure from councils who are outraged by its three waters decision.