Not a lawyer - sharpened teeth and elbows with lass with acquired brain damage- most is opinion born of experience.
Sorry DD's unwell. If the situation with GP hasn't improved you might want to send this to the GP and their practice manager: https://www.england.nhs.uk/wp-content/uploads/2019/10/improving-identification-of-people-with-a-learning-disability-guidance-for-general-practice.pdf
They would struggle to show how they were meeting their duties to DD under the equality act if they have no LD/vulnerable patient register, or equivalent.
Quite a few different ways of looking at hospital DOL depending on if it's physical or mental health issues that have resulted in emergency admission.
IME the DOL is likely to be an 'Urgent Authorization' by the hospital themselves which allows automatic seven day DOL, which they can extend by another seven, while a standard authorization is sought. It would be applied as DD already there, needing treatment.
It's so that hospital staff can legally refuse to open a locked ward door for her to take herself out and about unaccompanied, ie to the cafe, hospital shop, or out and about outside the hospital, or discharge herself if she wanted to.
When they make it they should apply for a Standard Authorization at the same time, and must have 'reasonable belief' that all 'standard authorization requirements are likely to be met.'
It doesn't affect physical treatment decisions, and in a good setting should do absolutely nothing more than keep her safe.
It shouldn't affect MH treatment decisions, but anecdotally it sometimes appears to.
Where there are already poor quality care standards either generally or with specific staff, it can affect attitudes. This is where on physical health wards visible presence of cards/flowers and most of all visitors, can be needed to 're balance' those attitudes.
If DD's been taken in under MH issues, then I'd suggest talking to other parents of young adults both with and without SN's, on other boards here, as well as possibly Mencap, and Family Rights Group, because it's another and tbh not always great world. The MH act vs DOLs can get a bit complicated and aren't always used correctly.
A DOL authorization can't conflict with a valid decision by someone with LPA or deputyship, and that's why traditionally parents go after these things, especially if the DC has property or compensation payments that A.N.other's interested in acquiring.
IME your rights or assumption of rights tend to mean more if she lives with you.
If you are registered as DD's carer, then you you already have some additional rights around access, and consultation especially re discharge plans. If you are an unpaid carer then familiarizing yourself with newer rights granted under the 2022 Health and Care Act would be generally sensible.
Becoming and being a deputy isn't that cheap, and it's an annual fee as well as some ongoing work from yourself. But if there's ever likely to be major differences of opinion between you and any form of care provider, then IMO it is worth applying for it. (For what it's worth I didn't use a solicitor, just presented facts and reasoning's to the court of protection - LA presented their objection and desires - court decided in our favor and awarded, but things change all the time.)
If you are already an appointee for your DD and she receives PIP (possibly the same for UC) remember to let them know she's gone into hospital as a change of circumstance (easy to forget in the thick of things. It only affects her payments if she's in for 28 days or more, or comes out but returns as an inpatient for the same condition within 28 days, but failure to declare even when payments are unaffected, can cause issues. If you're not an appointee and she's not in a fit state, notify as her 'carer.' (If they say you can't, don't argue just state 'this is a record made of the attempt to notify change of circumstances on behalf of X while they where unable to do this themselves. Thank you for your time. Goodbye.)
Hope whatever's happened is able to be satisfactorily resolved soon and normality resumed.