A child can have a disability but not be diagnosed. I think you are wrongly equating having a disability to mean having a diagnosis, which isn't the case. You don't need a diagnosis to have a disability.
The Equality Act "defines a disabled person as a person with a disability. A person has a disability for the purposes of the Act if he or she has a physical or mental impairment and the impairment has a substantial and long-term adverse effect on his or her ability to carry out normal day-to-day activities (S6(1))."
The guidance states "It is not necessary for the cause of the impairment to be established, nor does the impairment have to be the result of an illness."
You don't need to be on a waiting list for diagnosis, you could, in theory, apply without ever pursuing a diagnosis. It is reasonably common with ASD for parents not to want to seek a diagnosis for their child, but the child still has all the same difficulties they would have if they were diagnosed. They could still claim DLA. To use another example, a child of a friend has muscle wasting, HCPs aren't sure why, no diagnosis, but they have a significant mobility impairment and get HRC/HRM.
However, what you do need is evidence. In order to evidence a child's needs you are likely to need evidence from outside agencies, but that could be evidence focusing on a child's needs rather than aimed at getting a diagnosis.