We generally just say "dishes," I can be confident of that in American English at least. I think, but am not sure, the British English equivalent would be "china." In America we only say "china" to refer to a specific kind of dishware (fancy porcelain and fake porcelain enamel-y stuff) and we are probably more likely to say it in the noun phrase "fine china" than isolated as simply "china." Maybe посуда carries enough of a separate meaning in the Russian to warrant a translation beyond simply saying dishes or china. Although in English, so too does "dishes" in the sense of for instance "I'm washing the dishes" or "take the dishes out of the dishwasher." Because a dish can be a word that is a synonym for a plate or small plate, specifically that exact type, or it can also refer to all sorts of dishware generally. So when we say "I'm washing the dishes" it is an added meaning; we're not saying we're literally just washing only dishes (plates and saucers) but any or even all of our dishes, and in the case of most speakers they're probably just as likely talking about their utensils and cups and glasses as well--pretty much all the stuff they eat and drink with. Also I think saying "dishware" is more common than tableware or dinnerware, although all three are not the most natural in conversational English. But even if dishware is not more common, it has got to be at least valid so I would recommend adding it to the accepted answers. I was marked wrong for "Cups and plates are dishware."
I don't ever recall hearing "dishware" before in Scotland. Does that include things like knives and forks? Because those would be included in посуда.
You are wrong about the meaning of "china" in British English, that would refer to your good quality dishes which probably are made of china. Everyday cups, plates, spoons wouldn't be called china.
Among English speakers in Quebec (there's enough variance within Canada that I don't even feel I can say "In Canada" for these things), "dinnerware" is a completely foreign word, and "tableware" is not far from dinnerware too. Unless referring to a specific item (cup, plate, fork, glass, etc), it's just "dishes".
No, that's not true. "Tableware" would be totally acceptable in a corporate / industrial context, though it would be really weird to say in most normal domestic contexts.
(what I mean is "cups and plates are tableware" would be fine in the context of, for example, one inventory control programmer talking to another about how to categorize items)