As in there was a pile of washing-up in the sink patiently awaiting attention. I can’t recall whether I’ve ever heard this term in a US context. I think I’ve heard Americans say, “I’m going to wash up” with the meaning “I’m going to have a wash, wash myself”. Here it would be assumed you were off to do the dishes if you said that. (Incredibly OED’s first cite for washing-up as a noun with the sense “table utensils awaiting washing up” is 1972! They really couldn’t find anything earlier?)

In my experience, an American might refer to the act as washing up (e.g., “Let’s leave the washing up for later”, though “cleaning up” would be more usual) but wouldn’t likely refer to the dirty dishes themselves that way (your “pile of washing-up in the sink").

I concur, as usual, with the good Doctor T.

As to washing up referring to washing oneself, as opposed to washing dishes, it can mean either in US dialect. The reference is determined by context. If you’re called to the dinner table and say, “I’m going to wash up,” it would be taken as meaning you were going to wash your hands and will be at the table in a few minutes. If you were getting up from the table after the meal and say, “I’m going to wash up,” it would be taken as meaning you are going to clean the dishes.

(Incredibly OED’s first cite for washing-up as a noun with the sense “table utensils awaiting washing up” is 1972! They really couldn’t find anything earlier?)

Undoubtedly due to the date of the entry, which was written in the 1920s, with some additional senses added with the 1970s supplements. I’m sure when the editors get around to the full revision, with the internet now at their disposal, antedates will be easily found.

It’s used in Australia too.

cf laundry meaning dirty clothes to be cleaned.

That sense of laundry is used in the US too.  It can also mean clothes that have been laundered and are awaiting the next step (pressing, folding, hanging, what have you) before being put away.