I'm so sorry you've had that experience - a friend of mine's relative had similar and it was terrifying. Not around allergies, but around other issues that were labelled by ONE doctor as "fabricated illness" which then apparently invalidated all previous doctors' actual diagnoses, despite the baby having been treated proactively at top London hospitals. Nasty situation.
However, IF you give your baby an epipen and it goes wrong, you will be in far worse condition than if you take him to be tested now - you can NOT do that.
You must take all your older child's paperwork with you to the GP, a different one if necessary, explain the family history and the reactions you have seen, and get him diagnosed through the NHS. I agree that private testing is not the right way to go as the NHS won't have any record of it and may not agree with its validity.
Also, the nurse who disclaimed any knowledge of your previous paperwork - did you talk to the GP Practice about that? Not necessarily as a complaint, although it would be valid, but her response started the cascade of you being disbelieved and therefore a lot of what happened to you could be laid at her door.
Did you ever find out who reported you in the first place? Was it a family member?
But you MUST MUST MUST get your baby diagnosed before you attempt to give an epipen, you really must. Giving him an epipen NOT prescribed for him is assault, even if it's lifesaving.
Go back to the doctor. Please.