I have fought and won a personal injury claim. It was very, very stressful and took several years although I would very much doubt that would be the case here.
You have, I believe, three years in which to make a claim. At present it is impossible to know if there are any long term implications, the extent of the scarring or for how long your son is in pain or inconvenienced.
Keep record of appointments, all and any medical letters and receipts for any costs at all including for example, hospital parking, additional taxis, wheelchair hire.
Then later, when the dust has settled consult an experienced and reputable lawyer and ask the question. They will consider if there is a claim and place a value on it. It will be more complicated than you expect.
The issues you have regarding pasterol care and medical care onsite are rather separate in my opinion and a letter to the head teacher and governors asking how this was allowed to happen and stating your dissatisfaction with the care provided should be written straight away. Without mentioning any legal action at all.
I hope your son recovers quickly from such an upsetting and painful injury. And yes, of course compensation exists for pain and suffering (psla: pain, suffering, loss of amenity - unfortunately it tends to be low amounts even for life changing injuries ).