It's all to do with the child being in nursery on 'headcount day' this is when the nursery list all of the children they have eligible for funding on that day. The local authority use this list to work out how much to pay each nursery. If you don't get your child into the setting on time the nursery might not receive the money so will be reluctant to take him. If you can get him enroled in time then do, if not, try getting the nursery to speak to whoever in the local authority administers the free funding scheme as they may be able to help/make some adjustments if they know your child is moving setting in advance.
27th September is, I think, the headcount date for Early Years funding. Contact the EY team in the local authority that you are moving to- some areas are able to make exceptions for children moving into their area. If they don't, then the next headcount date is in January so you would have to pay full fees until then.