Hi @OneCoolGuide
As PPs have said above, firstly I would suggest your cousin got in contact with social services and spoke with them directly, as they will be able to offer much more support and specialist advice.
From my understanding, relinquishment is not necessarily an easy process. Social services will want to investigate all possible family options for adoption, so it might also invove seeing whether extended family of either birth mum or birth dad would consider raising the child, whether birth mum could be given extra support, or whether birth father or his family could take care of the child. Has birth mum considered this option? Regardless, this process. will all take some time to be done properly. If she refuses to care for the child in the interim, it may well be that the child goes into foster care in the interim, and if adopted then they will have to go into foster care whilst proceedings happen I should imagine.
Just so you know, very few children are relinquished nowadays, so there is maybe not as much understanding about what this would entail in the general public. The care of children would be relinquished to the state (I.e. foster carers) who would then seek to place the child with parents that the state chose. this is because the legal relationship between the child and the birth mother is severed permanently. If the child has very severe and complex disabilities, it may be trickier finding them a permanent placement, and so there is a chance they could be placed in long term foster care as it may be better suited to their needs, but it would be up to the local authority to do that. From my understanding, even if the birth parent relinquishes the child, they do not get a say in who the adoptive parents are, unlike in some parts of the USA. They are able to state a preference about things like religion, but ultimately if the state deems a prospective adopter a good fit, they will place the child with them. The birth mum may not know anything at all about the adoptive family, depending on the situation.
Modern adoption in the UK is always a last resort, and quite rightly great pains are taken to try and support the family to keep the child within the birth family wherever possible.
I'm really sorry to hear that your cousin is struggling. I wonder whether you could support her to reach out to some of her support network and get some help? Being a new mum can be horribly overwhelming, and with 4 children I can imagine it could feel quite hard to manage. However, and I speak as a mum of adopted children that I love and adore, adoption should always be a last choice, when all other safe options have been exhausted. I can imagine that in the throes of early motherhood, parenting a child with disabilities could seem very daunting, but if there is help available that would support her rather than going down this route, it is often better for the child in question.