Surely hiring a nanny doesn't mean the family has to suddenly become very clean people.
It doesn't sound like they live in squalor, it sounds like they're just normal messy. Few plates and cups on the side with some crumbs from breakfast. Sticky floors because, well, kids can do that. Stains on cupboards, pretty normal unless excessive.
I'm not saying people need to live this way, you sound like a clean person so I can understand why you keep your house very clean.
But the family have employed you to help care for their children. They are living a moderately messy life. If you don't like it, leave. But getting childcare doesn't mean the parents suddenly have an extra hour a day to clean up.
I have three young kids and we both work full time. I also have a small pile of breakfast stuff on the side each day, waiting for dinner time when I do the dishwasher. There's crumbs under their chairs sometimes when I can't be assed to sweep for the 3rd time that day. And I have piles of washing that needs folding and putting away. I could be a lot cleaner, but I'm already juggling working and three kids so I've lowered my standards.
You're a part time nanny. Of course you have more time to clean than working parents. Give them a little slack. And if you can't, be gracious and leave. You don't need to embarrass them.