I am assuming she has a Bangladeshi Passport because she has a Share Code.
If she has lived here for 5 years she is possibly eligible for ILR.
She can go down to the Bangladeshi High Commission in London and get a replacement passport, emergency travel documentation and a replacement NID. She will need to make an appointment and find out what documents she needs to take. They will accept the application because social services have been involved and there is a safe guarding concern.
She will need to report her passport stolen and get a crime reference number. She needs to state that her passport is being withheld unlawfully by her parents and she wants to report the documents as stolen. She can say she has fear for her personal safety and would like them to accompany her to retrieve the documents. This would save a lot of money but also put her through the trauma of facing them. If any of this is refused, ask for a supervisor and escalate it. Even raise a formal complaint if you have to. They need to issue a crime reference number at the start.
Her case with child services needs to be reopened. She might be physically safe but she cannot access employment, education, financial services or health care. They should be made aware that the parents are refusing to give over the ID and should help intervene.
I am not sure why you are being asked for ID to register with a GP. Surely, you simply need to be in the catchment area to enrol at the surgery. What specifically is the reasons he is being declined.
For a Bank Account, Try Monzo 16-17 account. They will accept her college offer letter. The Share Code. They can do alternative ID checks.
The DWP can issue her with a national insurance number with her share code, college letter and a letter from her child services case worker. She wont be able to upload photos but she can still apply online and then be seen in person at an appointment.
Once she has her passport - she needs to apply for ILR and get help with those fees through either social services or Childrens charities. She needs to do that because her next step will be applying for her British Citizenship.
Good Luck.