Law enforcement agencies were not as well networked as they are now. My experience is that whether a 20 year old recod shows up is a crap shoot. In most cases they do,. But there is always the odd case where an older record, particualrly one from another state, does not show up.
I agree with the first answer with respect to why your DUI did not show up.
As far as explaining why you did not admit your DUI the first time, I am confused. If you are applying for a new job with a new company why would they know or care about your application to the other company? If you are applying for a new job within the same company, then your concern makes sense.
If you are applying for a new job in the same company, I suggest you admit the mistake upfront and immediately. Most companies are going to respect your honesty and, in my experience, are more upset when you do not admit to your record then the fact you have one. And if you made a genuine mistake, then they should take into consideration your employment record to date, as well as the age of the DUI, in considering whether to take action against you. I think odds are against you being fired.