No, there's no duty to admit a disabled child if a school is full unless they have an EHCP (and that only applies to state mainstreams, and most special schools - a private mainstream can't be made to take any child at all, but must give reasons if the child is disabled and they say no). That's not unfair discrimination, because they're not treating that child any worse than any other kid. If a school is full, then they say no to everyone.
I think there's a misunderstanding on what the Equality Act can do. It doesn't say you have to treat a person with a protected characteristic better than anyone else, just that you can't treat them worse. So a school can't refuse to admit a disabled child on the grounds of disability unless they can clearly reason that there are no adjustments they could make - but they can truthfully refuse to accept any more children if they're full, and then the disability isn't an issue because they would say no to any new applicants.
The issue with the OP is that they don't want her child because she mentioned that there may be a neurodiversity, when they would have accepted a child who didn't. There was no attempt to establish what additional support that child may or may not need, and if they could meet the needs really easily, which would often be the case given most disabled kids never come close to needing an EHCP. That's unlawful. But if they had no spaces for any kids, and just said so and put the child on a waiting list like anyone else, that would be fine, because that's not worse treatment due to disability.
@NewName24 same issue with 1:1 in schools, agreed. I was pretty startled that my friend's child had that level of support at pre-school, no EHCP, as once in school, it would be (and was) a lot harder.
EHCPs offer protection in funding terms, if the parents are aware of the legal position, because if LAs don't adequately fund provision in an EHCP for a school, then the parent can take action against the LA, to force them to do so. It's going to be a lot trickier if the funding isn't adequate and it's via an EYFS support pot as nobody has the power to make the LA compensate properly. I was very surprised that this child's nursery were able to sort it for him in the years before he had an EHCP (that came once in school), but clearly they did.
None of the above, though, changes the law, which is that no business may discriminate against a disabled person, unless they can demonstrate that they can't make reasonable adjustments that would enable a disabled person to access the provision or service. Banning disabled people full stop, as this nursery is apparently doing, is not lawful.