Will the Police National Computer still hold information on drunk and disorderly convictions over 15 years old, that were committed by someone under 18 years old at the time of the offence?
Probably, but what you want to know is "Can anyone find out". It may not be there as some forces never bothered to record this type of offence. Why not ask the Police for a copy of your Data Access record to make sure. It might cost a tenner or so but it'll deal with your query.
There are ACPO guidelines for 'weeding' which recommend certain offences be removed from the PNC after a certain period of time. For example, someone gets done for Drink Driving and doesn't re-offend for 10 years. The ACPO website gives more details and you can always ring the Data Protection Officer for your local Police Force. Not 100% sure about how this works and not even sure if all Forces adhere to the ACPO guidelines but it's possible that if you pay for a printout, it won't have anything on there.
Check with your local Citizen's Advice Bureau about "Rehabilitation of Offenders". They should be able to tell you what are considered "spent" convictions. I am pretty sure that in the case you describe the offence would have been "spent" a long time ago. The length of rehabilitation usually depends on the type and length of the sentence and the age of the offender at the time.
The ROO Act does not wipe out the convictions recorded on the Police National Computer, it just details when convictions are deemed never to have been committed. For some jobs, like taxi driving, they must declared even if spent.