Hi disneystar - I have never heard of a public patient having a private anaesthetist, and even if you could find one to be on call for you, they still charge the private patients about £600 in our hospital.
If there are no available midwives to help doing an epidural due to emergencies etc, then they cant just suddenly become available because of the private anaesthetist IYSWIM, so you still would not get your epidural.
The midwives in most NHS hospitals, who care for private patients are public midwives, there are no 'private' midwives looking after private patients in labour wards, so they are just as busy and have to just split up their workload as best they can.
If you go private, most hospitals charge from about £5-10,000 for private care, so unless you are loaded, it becomes a very expensive option just to get an epidural.
Most of the time, most people will get an epidural within about 20-30 mins, but sometimes it just happens that the unpredictable nature of labour ward will not allow this. But it is unusual not to get an epidural when you ask.
Sorry that this is not what you want to hear, I suggest that you speak again to the midwife in your hospital who suggested this and ask her to investigate it.