Generally, schools have a 'no hat, no play' policy through the warmer months. Children must be wearing their hats whenever outside, otherwise they must play in the shade. Some classrooms have a sunscreen pump, but it's not mandatory. If you want your DD to reapply at lunch then perhaps show her how to do it herself and send her to school with a tube. You could also encourage her to play in the shade on extreme days. Schools are very good at teaching sun safety so I don't really think you need to worry. As for swimming, if it's outdoors (I've only known school swimming to be indoors), I am sure the teachers will be right on to sun protection, but send your DD with sunscreen, a hat and protective clothing anyway.
Remember, a little bit of sun each day is very good for children too. Vitamin D is crucial for brain and body development. With their hats and uniforms on, only a small amount of skin would likely be exposed in the short time that they are out at play anyway. This will likely do more good than harm.