dd2 was in a private nursery. if the private nursery allows you to use your nursery education grant from the term after three (even if they require you to top up) they have exactly the same access to the lea early years service, including assessments from the area inclusion officer, access to ed psychs etc etc etc ad infinitum.
unless it is a private nursery attached to a private school, and there is no link to either ofsted inspections or whatever, then there is absolutely no reason to move. just ask them to utilise their lea contacts and improve their relationship with the sn team.
sometimes nurseries are just a bit clueless and need a bit of a prod, esp with sn. you'll be doing them a favour really, and then they'll know what to do next time they come across a child with sn. looking for a mythical 'sn ready' nursery place is a bit like looking for a needly in the haystack. if you come across one, it's more by luck 
just read up on ieps. read up on school action and school action plus. and then ask to discuss them with the senco. and then ask if they think it would be agood idea to get the area inclusion officer to come in and assess.
or just call the area inclusion officer at the lea yourself (just ask for the early years team and ask who deals with x nursery - it's very easy) and ask them lots of questions about early years and appropriate support. and then go back to the nursery with the info.
unfortunately, an awful lot of the sn road involves doing the mileage yourself as there's no one road map that suits every child, and tbh, no standardised destination!
but ultimately, if it is a private nursery attached to a private school and is not ofsted inspected or eligible for nursery education grant, you probably will be better off moving. even if it is to a different private nursery. getting a council place is usually impossible in most areas. (we managed it for a year from dd2's first birthday, but she was in receipt of high rate dla and we had no family support and i had two other pre-schoolers and a dh working away... the sn hv had to refer us to the panel. dd2 got 2 mornings a week) we then moved out of the area, and moved to a private nursery where the lea paid for 5 sessions of 1-1 care.