You can't undo his sunburn, sorry but you can't.
A child has been burned at nursery, have they put it in the accident book? They have to, they would if it was a burn from a kettle of water, this is no different, it is something that should have been prevented.
They have also not given first aid to a child with a burn, how can they excuse this?
Once it is in the book, and if necessary insist on seeing the book and writing it in yourself, if they won't show you it then there are steps to take.
They then need to take any steps necessary to prevent this happening again. So a policy of all children wearing hats when outside, regardless of weather - odiously summer and winter hats will be different.
Staff training - staff must have appropriate training to stop this happening to another child. You can burn when it is cloudy, through clothes, through windows, black skin can burn - I know I've done it all except the last one.
Staff training on what to do if a child does show signs of sunburn and or heatstroke.
Not an expert but don't nurseries have to comply with certain rules? Be registered with the local authority? You can contact them with concerns. Also the health and safety executive.
You can get various UV monitors, some are electronic devices others are more simple and can be put on a child's wrist - note I have no idea how accurate they are but a nursery could easily buy a box so each child has their own.
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OK so that's what they need to do in the future, you are also entitled to some compensation from them, they will have insurance, you need to get this logged with their insurance just in case (and I am not for one minute wanting to worry you, but this has to be said) he does go on to develop skin cancer, if that does happen then the insurance will need to pay out much more than they would for a burn.
This is complicated and I think you may need legal advice, personally I pay for legal advice with my contents insurance, you may have that, or your son may be entitled to legal aid if you want to go down this route.
He also needs a trip to the doctor to have the burning noted.
When you talk to the nursery do not ay 'sun burn', say burn.
The bottom line is your son got extensive burns in their nursery and they did a) nothing to prevent it b) not put it in the accident book and c) did not give first aid treatment
keep remembering that and keep saying it to them.