OP, I understand how worrying this must be for you, but there is plenty you can do to help. Both my youngest DD and I use blue Ventolin inhalers, and she has needed to have an inhaler with her at all times since she was 4, so I have lots of practice with this particular issue!
If you need an inhaler as a matter of urgency and can't wait for the prescription, go into a Pharmacy and explain the situation and tell them you have requested the prescription and will bring it in for them when it arrives, but you need the inhaler now and they will usually let you have one.
I would request 3 more inhalers from the Surgery after the prescription for the one you have just requested arrives. I keep them everywhere - I've never thought "Gosh, how annoying to have so many inhalers everywhere", but I'd still rather that than "Gosh, my child is having an asthma attack and I can't find an inhaler". I have one in my car, my handbag, my bedside table, DD's school bag, DD's school office, DD's bedside table and 2 spares in my medical cabinet. Obviously you need to keep track of the use by dates on them, but I'd rather be safe than sorry.
My DD and I both find the cold air triggers difficulty with our breathing and find that wearing a snood type mask helps to warm the air before we breathe it in, so might be worth trying with your little one. I really hope you manage to find a comfortable solution for your DS and that it is able to put your mind at ease whilst he's at school. 