Hello, thank you for your question. My name is ***** ***** I can assist with this.
Firstly, I'd like to think that the search result from 30 or so years ago would have no bearing now, particularly when you were 14 years old when the cautions were given.
What were they for?
Okay. Regrettably, because there are two cautions, these will not get removed on an enhanced check. New rules came into force last year which allow an old conviction or caution to be removed, but it must be the sole conviction/caution. As you have two, these will always remain under the present system. The details of those rules are here: https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/266493/Filtering_rules_for_criminal_record_check_certificates_v2.2.pdf
I wouldn't have thought so, you've been working for the NHS for 14 years without issue. I doubt they would be interested in knowing what you got up to at 14 year old, almost 30 years or so ago!
I understand why you might be concerned, but I don't think you need to worry.