Yeah, if you’ve got no internet then you can’t do all the nice internet things.
Pretty sure you’d need internet (wifi would do) to set up Apple/Google Pay on your phone, but I don’t think you need internet when out and about paying for things.
Looks like the Google one needs to phone home every so often, but should work for “a limited number of transactions”. Whether that’s by value or by transaction count I don’t know. It certainly seems to be instant whenever I use it, which makes me think it doesn’t rely on internet access.