Not a lot. The disability register has 100% changed from what it used to be, and "registered disabled" is now meaningless.
Once upon a time (when my grandmother got various benefits) the council held a thing called the "disability register" and in order to get DLA or similar, you went to your GP, the GP got someone to examine you, and on the basis of that examination the council decided whether to put you on the disability register and what benefits you were entitled to.
A while ago, the govt decided to centralise this, and in particular make all the DLA decisions centrally. So the key benefit is DLA and the main "sign" of being disabled is whether you get DLA or not which is now done in central government offices and nowt to do with the Council. Things like blue badges etc. and other things like eg getting a carers' pass at legoland are now most often based on production of a DLA letter. Some places still say "on the disability register" but it's mostly just they don't realise it's obsolete.
Many councils still keep a "disability register" but it is more for their own records: allows them to be comfortable their Social Workers and Health people are all aware of how many disabled people they have and where, and also it allows them to administer minor local schemes like cinema passes. But it's entirely optional and there are plenty of people in receipt of DLA not on any "disability register" and I think quite possibly councils that don't formally have one single register any more.
The other change is that neither DLA nor the register convey legal rights, because the test of legal disability seems to be a common-sense one: ie if you are disabled from doing day-to-day things, you are disabled, regardless of whether you fill a form or appear on a list. Your son has rights by virtue of his disability. Your husband has rights as a carer at work by virtue of your son's disability. Though being on some register or, more powerfully, in receipt of DLA, will make his disability more credible, really all your husband has to do is to make it clear to his HR dept that he is caring for a disabled child and that is sufficient.
Ditto statement - though medical diagnoses are helpful to get statements, if the child's education is impacted then they are less able - disabled - educationally and you should be able to apply for support, potentially to the extent of a statement based on the result of his condition, ie how it affects him, without needing to prove the cause.