I never class myself as being a "cloak room lawyer", as I try to offer advice from the emotional behaviour aspects of humanity.......
You do not say what the offence was - in ways that does not matter as long it is not one that should bar you from the work - and I would suggest you must have a constructive working relationship with a superior in the Police.
I would again suggest that that person will find out anyway so the positive stance I would take is to approach that person, explain what you can remember and ask for their support/advice and how to complete the form. Remember you have worked there for 15 years so i would assume you have had good feedback on reviews, so use them as references to demonstrate your rehabilitation.
You have shown your courage and determination not to re-offend so carry that with you. If unfortunately you can no longer carry on with that job do not be sad. I am sure you will get a written character support that states your true character but unfortunately the law is the law and so that means you can no longer carry on.
Good luck and I hope this helps and they believe in rehabilitation.
Learn from yesterday, live for today, hope is for tomorrow else what is left if you remove a mans hope.