My experience of it wouldn't be like a hangover. As well as nausea, people who feel that ill with a hangover tend to have headache and sound/light sensitivity.
While I do have nausea with "poisoned" there's rarely a sense I might actually throw up, which isn't always the case with a hangover. It's just grinding nausea.
The nearest thing I've experienced is being shut in an enclosed space with wet gloss paint. The fumes seem to get right into your mouth and nose and lungs, so you wish you could sit in a stiff onshore breeze with a pint of chilled water to wash it out.
"Poisoned" is similar in that something has got right inside your body, but it feels as if it's in your bloodstream and gut as well as mouth and lungs. You're just as desperate to get rid of it, but unlike getting fumes out of your body, you can't imagine a remedy. I find myself drinking a lot of water, but it doesn't help.
I think if you'd experienced it, you'd know. The only adequate word for it is "poisoned"; it's striking how many people come up with the same term.