Owners Yorkshire Building Society (YBS), the UK's second largest mutual, said it would close 28 N&P branches and 20 YBS branches this year.
Customers who want to continue operating a current account will have to find a new bank or building society.
A total of 440 jobs are at risk as a result of the overhaul.
The N&P customers will be contacted and told of their options before their accounts are closed on 31 August 2017, as the YBS concentrates on savings and mortgages.
The announcement comes a year after YBS embarked on a separate rebranding and branch closure programme, in which N&P was unaffected.
On Tuesday, HSBC said it was closing an additional 62 branches this year owing to the growth of mobile and internet banking.
YBS has grown strongly in the past decade after merging with several other societies following the financial crisis of 2008 and 2009.
It took over the Barnsley building society in 2008, the Chelsea building society in 2010, and then the N&P in 2011.
The N&P's finances had been damaged by a huge Â£51m compensation bill, after the society was found to have mis-sold high-risk investment polices to 3,300 of its customers on behalf of the now defunct investment firm Keydata.
The Barnsley and the Chelsea were rebranded under the YBS name last year and now the same will happen to the N&P.
"We continually review all areas of the business to ensure we remain focused on what our members require while utilising their money as effectively as possible," said Mike Regnier, chief executive of YBS.
"The driving forces behind the proposed branch closures are shifts in market conditions and an increasing desire among customers to transact digitally rather than on the High Street.
"We therefore no longer believe it is the right commercial position for us to continue to maintain these 48 branches across the N&P and Yorkshire network."
All current accounts with the Norwich and Peterborough building society will be closed, the brand abolished, and some branches closed.