my child is at secondary school so we have been through this. I'm afraid most schools do expect the child to carry their adrenaline and adminster it themselves. However it is not true that teachers can not adminster adrenaline. They cannot be REQUIRED to do so but in most schools the education authority invite them to volunteer and provide insurance cover for those who do so. Also all schools are required to have first aid cover. The office staff are often trained and in a large school there should be several trained people. The first aiders CAN be required to adminster medication as part of their contract.
At my child's school there are several teachers who volunteered to be trained. I attended the training session myself taking an extra practise pen and out of date epipens. The school nurse did the training and showed a video. She had a trainer pen but was less proactive about getting people to try it About 10 - 12 people attended the training session.
I had raised the issue of nut allergy with schools before deciding what school my child would attend and chose this school partly because they seemed most clued up. I also checked, before my child started, when they last had any training ( it had been a couple of years) so I spoke to the school nurse to enlist her support in making sure they did update their training. As yours is obviously not helpful I'd check with the LEA what their policy is on ensuring teachers. Do you want to say which LEA it is?
It sounds as though you may need to enlist outside help in the form of the anaphylaxis campaign. If you aren't a memeber you can still ask for help. They have regional volunteers who may help you negotiate with schools - I was offered that help. You are posibly going to have to involve governors too.
At 11 your child does also need to start learning to self administer and she can practise with an out of date pen on an orange or apple. It is also useful to have a buddy system. Children are less upset by the idea of using a pen and are likely to be with your child often.
My child carries 2 epipens with her and has one in a clearly labelled box with first aid staff. I have spoken to staff in the canteen about the food. It's all supposed to be nut free but packets of nuts and seeds were on sale briefly. They proved to be pine nuts but your school may sell nuts. In your case I think I'd want 2 with the child and 2 with the first aider.
You are welcome to contact me if you think I might help further and I'll try and remeber to check more often than usual.
I don't want to worry you more but it is not unknown for children to throw nuts at a peanut allergic child "to see how bad their allergy was". You have to be very assertive and persistent if you want to reduce the risks for your child. Also first aider go to lunch. there is supposed to be cover at all times but if you go into the school you'll find that doesn't always happen. That's why I say a buddy system can be useful.