they can definitely lawfully refuse to admit them.
No, they can’t.
Without an EHCP, state school places are allocated in accordance with the oversubscription criteria which must comply with the admissions code. See this IPSEA page which states “A school must not refuse to admit a child with SEN because it does not feel able to cater for those needs.” [their bold] A school cannot lawfully refuse a place based on a child’s lack of toilet training.
With an EHCP, for non-wholly independent schools, the LA must name the parent’s preferred placement unless the LA can prove:
“a) the school or other institution requested is unsuitable for the age, ability, aptitude or special educational needs of the child or young person concerned, or
b) the attendance of the child or young person at the requested school or other institution would be incompatible with—
(i) the provision of efficient education for others, or
(ii) the efficient use of resources.”
This is set out in section 39(4) of the Children and Families Act 2014. If the LA cannot meet the high bar to prove one of the above, the LA can and must name the preferred placement even if the school objects when consulted. Not being toilet trained would not meet this legal threshold.