The Metropolitan Police were called to the shop in St Nicholas Avenue at 6.49am on December 30.
A 52-year-old man had reportedly threatened staff with a knife.
He was detained by staff with the help of a member of the public before police arrived, but suffered injuries during the struggle and paramedics were called.
Fighting for his life, the man was taken to hospital by the London Ambulance Service.
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The man died in hospital on January 1.
The Met says his next of kin has been informed and are being supported by specialist officers.
A spokesman said a “special” post mortem will take place in due course.
