Mr Garner will join Nationwide in spring 2016 to replace current chief executive Graham Beale, who is retiring.
He will leave Openreach, the fixed-line network provider, after almost two years in charge.
Mr Garner said the Nationwide job was "too good to turn down".
He said: "I've greatly enjoyed my time at Openreach and deciding to leave has been an extremely hard decision, but Nationwide is a firm I long-admired during my time in financial services."
Mr Garner was previously head of HSBC UK, where he worked with banking clients including First Direct and Marks and Spencer Bank.
Nationwide chairman David Roberts said: "Joe is an exceptional leader and is exactly the right person to lead the society through the next chapter of its journey."
Gavin Patterson, the BT chief executive, said: "I am sad to see Joe leave but he is moving to a terrific job and I wish him well."
His departure comes as communications regulator Ofcom investigates whether Openreach should be split off from BT.
Rival internet providers argue BT has an unfair competitive advantage from owning the telecoms infrastructure business.
Most internet service providers offer services through the pipes and cables run by Openreach.

Previously, trainee doctors had to wait up to four years to experience the consultation room.
GP Live was developed by Aberdeen University and uses streaming software to enter the consultation room.
Researchers hope it will encourage more students to join general practice and help ease a recruitment crisis.
The software will stream consultations just moments after they happen in an effort to give students a flavour of life as a GP.
The system utilises one of the University's Digitally Enhanced Learning Spaces (DELS), where students can engage in interactive learning with the aid of 55-inch display screens and state-of-the-art web conference tools.
It was developed by Dr John McKeown, a senior clinical lecturer and GP at Cults Medical Group in Aberdeen.
He said: "The biggest appeal for students is that they are seeing consultations that are almost live, and this adds a sense of immediacy that appeals to students who have grown up in an era where digitisation has made learning far more interactive.
"We aim to make the sessions as engaging as possible by discussing the GP's approach while the consultation takes place - for example how they communicate, the direction that they take the consultation, and of course their recommended course of action."
Dr McKeown said that one of the main benefits of the system was that students and lecturers would be able to discuss consultations that have taken place at a variety of practices, where the issues could be very different.
He added: "This means that students get a real insight into the challenges facing GPs who might operate in less affluent areas, or in remote rural locations.
"The system also allows us to work through a number of different consultations in succession, which very much mimics a typical morning for a GP.
"This allows us to see the range of skills - interpersonal, diagnostic and otherwise - that they bring to a normal working day."
Dr McKeown said that by the time current first year students graduated there would be a pressing need for new GPs in Scotland.

The incident on 23 April involved a Russian SU-27 Flanker fighter and a US intelligence-gathering RC-135U.
The US plane was flying a routine mission in international airspace over the Sea of Okhotsk, officials said.
The incident came amid tension between the US and Russia over Russia's involvement in Ukraine.
"The SU-27 approached the RC-135 and crossed the nose of the US aircraft," Lt Col Myles Caggins III told the BBC.
The Russian aircraft manoeuvred aggressively and exposed its belly to the American pilots to demonstrate it was armed with air-to-air missiles, Defense News and CNN reported.
US and Russian defence officials are said to have later discussed the incident.
Earlier in April, another Russian fighter jet made repeated low-altitude passes over US Navy destroyer Donald Cook in the Black Sea.
On Wednesday, US President Barack Obama condemned Russian "aggression" in Ukraine .
"We will not accept Russia's occupation of Crimea or its violations of Ukraine's sovereignty," Mr Obama said in Warsaw.
He and Russian President Vladimir Putin are expected to meet in France on Friday during 70th anniversary commemorations of the D-Day landings in Normandy.
Nationwide building society has appointed Joe Garner, currently boss of BT's Openreach unit, as its new chief executive.
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A new project has been developed to allow first-year medical student in Scotland to watch GP consultation for the first time.
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A Russian jet flew within 100ft (30m) of a US Air Force plane over international waters near Japan in April, US officials have said.