Shops in cities including London, Manchester, Newcastle and Cardiff were shut down or only accepted cash because of the glitch.
Hungry regulars were left in despair, with many sharing their dismay on social media.
Bal Bansal tweeted: "Just heard Greggs are having an IT issue and having to close shops.
"Personally, I think this is bordering a national emergency!
"Can all IT folk go and help Greggs...potentially, steak bakes as a reward!"
Another customer wrote: "Greggs this morning cash only! Sitting here with my coffee watching almost everyone have to walk out."
Unclear
Another tweeted: "Why are all of the stores in Newcastle city centre closed? While other closures were reported in Manchester, Cardiff and Westminster."
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