Is it normal to say, I have washing and washing-up waiting for me (to do).
You could say you have washing waiting for you to do. I would probably not say that, but it's fine. You can't have washing up waiting for you.
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You could say you have washing waiting for you to do. I would probably not say that, but it's fine. You can't have washing up waiting for you. I think it is because it is a verb. You would wait for something that is a noun, not a verb.
If you prepare a bathtub of warm water, you could say the bath is waiting for you.