I think it does depend from trust to trust, and even on who is in charge on the day. Generally they don’t like to give you one before you’re 4cm.
I chose my hospital deliberately because I knew they give epidurals easily on request, and also have a low threshold for intervention (very important for me; I’m pretty medical in my approach to birth). The anaesthetist confirmed to me that they have a target of siting an epidural within 25 minutes of being asked for one. BUT that’s once you’re on the labour ward. When I was admitted at 4am I was put on antenatal, as I wasn’t dilated almost at all, despite strong regular contractions. The (agency) midwife there made it very clear that I wasn’t going anywhere near the labour ward, or an epidural, until 4cm.
The pain became unreal by about 6am, and I spent the next 6 hours howling in agony. I’m not a screamer at all, but I found vocalising was the only way I could get through it. In that time I begged and begged and pleaded for an epidural but was told over and over again no way until 4cm. I was offered paracetamol (complete joke!), codeine, and pethidine, which I absolutely did not want, but took in desperation after being told again no epidural was coming any time soon. The pethidine did absolutely nothing for the pain, but scrambled my brain and left me with several memory gaps.
Someone, I don’t know who, took pity on me around noon, and at still less than 1cm moved me to the labour ward, gave me the epidural and broke my waters. I don’t know who it was, but I’ll be eternally grateful to them, because I honestly don’t know how much longer I could have gone on like that. It was by far the worst pain I could ever have imagined, and to this day I can’t find the words to describe it. And the epidural did me no harm, because I dilated to 10cm within the next 4-5 hours after 18 hours of almost no progress at all.