Request that the school share their risk assessment with you. It should be based on the student’s healthcare plan, but the risk assessment itself should not have any identifying information about the student. If it does, they can redact a copy for you.
It should contain information about the nature of the allergy, then control measures such as restricting access to the nut trees, especially during nut drop season.
For example
Remove fallen nuts from the ground promptly. Consider netting trees in nut drop season.
Train staff to recognise and respond to anaphylaxis and keep medical supplies to treat.
Train students not to pick, handle or share nuts. Students who want to can get involved with nut harvesting can come along to a supervised session (that you could run), with a parent or guardian outside of school time if desired.
Adapt lessons to avoid nut handling.
Fence them off and put up simple signs.
Remove trees only if risks cannot be adequately controlled. (Which would also need a risk assessment - and probably a professional contractor to do the work.)
From the school’s point of view, it may be a capacity issue. They have a parent who is desperately anxious and on the phone regularly - nut allergies can be terrifying. Staff may also have their concerns about if anything happens, as they are on the front line. No one in the school is a qualified allergist or otherwise confident enough to manage the risk - there might be insurance concerns in the mix. Keep talking to the school staff.
And if the trees have to go, be assured that you have still done a lovely thing for those students over the time you’ve been involved.