Yes. You can "prod" people to do many things.

It comes from the 1500s, meaning "to poke with a stick". It just means you are trying to make them to do something that they currently aren't doing.

I couldn't call it wrong, but it sounds odd to me.

Prod in this context is perfectly OK in BE I'd say

When I read the context, I also think "prod" would be used when the question had been answered in part already, but if the question hadn't been answered yet at all, then she would say, "You didn't answer the question," meaning something like:

"You didn't answer the question," she (purposely) pointed out.
"You didn't answer the question,", she said demonstratively.

I just don't see the "prodding" at that stage, since there has been no answer up until that point.

I'd go with prodded. I take Sproink's point, but would suggest that although his/her answer had not begun, the interaction was indeed underway - assuming the recipient had heard the question.

The recipient has heard the question, but hasn't given an answer yet, on purpose or not doesn't matter, I think.

Thanks a lot @all.