WILL is sometimes used to express a future action, but very often is not. Here are two of its most common uses:

1- We use WILL to express a command in a soft and polite way.
  - Open the door! (this command may sound strong and even rude)
  - Will you open the door? (this command sounds nice and polite)

2- We use WILL to express volition (what we want to do) and so its negative WON'T expresses refusal (what we don't want to do)
   - Will you show me how to cook this? (= do you want to show me?)
   - Wait! those bags look too heavy. I'll help you. (= I want to help you)
   - I know everybody's going to go, but I won't go to that party. I hate parties. (= I don't want to go, I refuse)

So the conversation in this little rhyme is:

- Will you get up?
- No, I won't get up.

It could be just talking about the future, but it's a mother and a daughter, so it's not just talking about the future. The mother is telling Mary to get up (command) and she refuses (volition).

Imagine this other situation, Mary is very ill and weak. In that case the mother is asking Mary if she feels strong enough to get up, and her question could either be "please, get up" (polite command) or volition (do you want to get up now?) or just a question about the future (I need to know if that is going to happen) .And Mary's answer could be refusal (I don't want to) or simply prediction (it won't happen now, I'm too weak).

So the meaning of WILL may vary depending on the context and the situation.