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Epidemiologists advising the NHS say that about 56% of the UK population - equating to about 80% of smartphone owners - need to use the app in order to suppress the virus.
However, they add that the spread of the disease could still be slowed even if the take-up is lower.
https://www.bbc.co.uk/news/health-52521526
(edit: shared for the simpler math more than anything, but I do think 80% isn't entirely beyond possibility, and I think the point about it not being a zero-sum game is important)
It's being reported that the smartphone app would need 60% of population to be using it to work.
So that's six sevenths (fuck that maths) of all smartphone owners using it. Yarp.