The tills have keys to manually open them, and you can just record transactions on pen-and-paper if needed and enter them later. I've seen plenty of businesses do exactly that during power outages. Not as fast. But totally doable.
Stores with tills and freezers etc will have power for the tills but the backing network for payments probably won't be up. That's the concern with being cashless. They can accept cashxl, but no one has any.
I was able to buy some groceries and pay with card. The tills had a battery backup and the network infrastructure that supports card payments was apparently working.
That said, lots of people hit the cafés and had to resource to cash payments. There was also lots of people buying bottled water at the shops.
So basically, you could divide people in two groups. Those that took it like an extra Sunday, and those that took it like the beginning of a war or something :-D
Most people around here seemed to be balanced. Yes, a lot of people outside at bars/cafes enjoying a spare Sunday, but also preparing quietly and slowly in case it takes longer time. All the stores ran out of batteries, radios and such, at least the ones we visited, and I only managed to make one card payment around 13:30 sometime, after that internet stopped working 100% until the night. Almost no super markets were open around here, maybe one or two who had generators it seems.