Smart meters do not use wifi
But the smart grid does. As I said, it's intended to be a very complex system.
This 2010 Ofgem document gives a brief idea of what the Home Area Network (HAN) and Wide Area Network (WAN) were planned to look like. Some of the communications in the diagram on p11 are by mobile phone, some are by WiFi and some may be by Zigbee. I've also heard more recent muttering about using power line communication (PLC) within the HAN, but I don't know state of play on that.
www.ofgem.gov.uk/ofgem-publications/63618/han-workshop-19-nov-ember-han-basicspdf
Quick guide to smart washing machines which use WiFi. The article doesn't mention switching by the power company, just switching by you, the appliance owner, but this is the sort of tech that is planned to be integrated with the price signalling: www.which.co.uk/reviews/washing-machines/article/smart-washing-machines-explained
There are different possible set-ups, but if I remember correctly, in the proposed UK system the smart meter does not send a signal directly to the washing machine.
IIRC, an approximate version would be: the smart meter sends a signal about how much power you're using right now to the Data Communications Co (DCC); as scaryteacher says, Capita have now been given the contract for this. The DCC sends that data to the power company and to a power company app or third-party app. The power company sends what the price rate is right now to the same app. Depending on what the price is and how much power you're using, the app then sends a signal to your home network, which sends a signal to your smart washing machine or smart socket to switch off or on.
That was off the top of my head, from memory, I haven't checked recent plans, and the plans have always been convoluted, so beware I may not have described the currently planned set-up exactly correctly. But you get the drift.
Each of those comms methods and nodes has its own security issues, of course, and there are significant interoperability challenges. There are also major risks associated with software updates. As consumers, we will find it much less robust than what we have now. Because right now I press a button on the front of my washing machine and it starts washing.