I recall a few articles over the last 2 years that stated that the waiting times as a private patient in Dublin public hospitals is as long as the waiting times as a public patient in the same hospitals for the same treatment. Unfortunately, I can't find a link to those articles anywhere, so I'm only going on memory.
Waiting times for private hospitals aren't published anywhere that I'm aware of, but my inclination is that access to private hospitals would generally be quicker than that for public hospitals.
Perhaps ask your insurer to give you an indication of the average waiting time for your local private hospitals? The waiting time for each type of treatment will be different, but at least with an average you have a better idea. Or even contact the hospital admissions office yourself to see what they say?