I am however also quite angry at the feeble excuse as to why bayley was left with baby wipes to cover her hand untill she arrived at hospital. I do understand what they said about how they don't usually have accidents like this and only see cuts scrapes and bruises. But for first aid trained staff to act in this way and not cover her hand properly is beyond me. And I will be suggesting a refresher course as part of first aid training covers bandages and burns.
I have been first aid trained for quite a number of years (the proper 3 day + defib course, not the 1 day), and actually only recently did the refresher as I always keep it up to date.
A child, with that injury? I think that depending on the mental state of the child I'd have been tempted to leave it open. It's quite a large area, wrapping with cling film will be quite awkward to do it effectively, add in that it is a child who isn't likely to give you their hand and willingly let you do it, where you are likely to cause a considerable amount of extra pain.
They are also not allowed to drive you to a hospital themselves, unless somebody there has the correct insurance, but, should I suppose could have called an ambulance, if there's no transport. At least them, keeping the child inside and still till a paramedic arrives, will help reduce the chance of infection etc.
But that is quite severe, and obviously needs medical attention. I certainly wouldn't be comfortable cleaning, definitely one for the professionals! Remember, that first aiders aren't professionals in the medical industry.
However, on to getting your complaint sorted, since your child received hospital treatment after the accident, then it is RIDDOR Reportable, which they should file to the HSE within 10 days of the incident.
The HSE will then perform their investigation into the incident, when you speak to them, you want confirmation that this has been reported to the HSE under the Riddor regulations. If it hasn't, then you really do need to be getting in touch with the HSE.