If your family had the means to pay for private assessments, these would be quicker than waiting for NHS assessment.
I think some specific legal advice would be helpful, because there are several issues intertwining (as I understand it, I may have got it wrong):
Your DN has developmental delays, but the underlying cause/a of this have not yet been identified.
Your DN has not so far been assessed by a paediatrician, speech therapist, occupational therapist or other medical professional (is that right?)
Your sister has a severe MH condition, has suffered a bereavement, tried to end her own life, and was admitted as a MH inpatient
SS have been involved, and the SW believes your DN's delays are the result of your sister's MH illness and her perceived inability to interact with her son to help him develop.
(Has the SW not recommended a supported nursery place to help with interaction and language / motor development?)
There is a strong family history on both sides of DN's family of autism and ADHD.
Your sister may herself be undiagnosed autistic.
The reason its important to look at all these different factors is because they all feed into the questions:
What does DN need, specifically, to help him develop in line with his full potential? (Ie if he's autistic then he's working to his full autistic potential and that will be a different trajectory to a non-autistic).
Which of DN's specific needs can your sister meet herself with her existing support network, and which can she not?
Of your DN's needs which she cannot currently meet, are there any adaptations that would help her to meet them?
If yes, how will SS /network provide these adaptations?
If not, what alternative arrangements would meet DN's needs, and support your sister to reach a place where she could then meet his needs?
I am not trying to be pedantic by setting it all out but I hope other posters can see that determining the underlying cause/s of DN's needs are essential to getting them correctly met.