Silver Voices, which undertook the survey of 650 members, said the app excludes the people most at risk from Covid-19.
The app is available to download for free from the Android Market and Apple App Store. However, the technology only works on phones which have iOS 13.5 or above installed, or Android 5.0.
Android 5 was released six years ago, and most Android smartphones are compatible with the NHS app.
However, iOS 13.5 was only released in May 2020 so requires users to have a very up-to-date operating system.
In addition, iOS 13 is only compatible on the iPhone 6s and newer so for many people with older phones it’s not possible to install the operating system.
Dennis Reed, director of Silver Voices, said: “The government has rolled out a Covid-19 contact tracing app that a large proportion of senior citizens can’t use – the very demographic most at risk from the virus.
“Policies should be geared to protecting those most at risk of dying from the disease, by comprehensive testing of those most in contact with them.
“The NHS Covid-19 app has the potential to be a vital source of information to slow down the spread of the disease, but it is failing the most vulnerable.
“We urge the government to update the app to ensure as many people as possible are able to install it.”