Rackmount atomic clocks still need a GPS signal today to be set or they have to be pre-set in a central location and then distributed.
Time syncing used to be a PITA and now it’s easy due to GPS so pretty much every protocol has a very narrow time tolerance. My work HP laptop drifts 3-5 min over the weekend of the VPN because it’s not set to an external time server and all hell breaks loose.
Yes civilization will not end over night but it would have a major impact we will find ways to get by but there will be havoc and the financial damages will be in the trillions heck the disruption to shipping and logistics alone would be immense nearly every one and every thing today is reliant on GPS either directly or indirectly.