I never had trouble getting an epidural in EITHER of my deliveries. In my first, I was induced and I was exhausted so I got it more for the exhaustion than for the pain. The pain was no where near unbearable. I asked for it once and they got it within half an hour or less.
In my 2nd, I was in more pain but what I found harder was that they INSISTED on having the continuous fetal monitoring because my daughter had a congenital condition (fair enough) - but due to that, I could only be in ONE position otherwise it wouldn't pick up her heartbeat. The pain wouldn't have been so bad, but not being able to move made it a LOT worse. I asked for an epidural and the nurse was dragging her feet. I started yelling at her (because it was becoming unbearable) and she managed to hurry it up a bit

I got it within an hour of originally asking for it - but I was doing a whole lot of yelling at her in that time

That particular nurse was awful though... thankfully her shift ended soon after and she was replaced with a nurse who was FAR nicer.
With my first, I had an episiotomy which was the most painful part of my recovery, but with my 2nd I decided to tear naturally and they allowed that, even though it was a vacuum-assisted delivery. I think that the vacuum caused me to tear more than I would have without it, but regardless, the healing process was much smoother than with my first. They DO listen to your requests at the public hospital, but you may need to be more demanding.