You've had a great deal to cope with of late, Sophie, and it's not your place to take the weight of the world your family on your shoulders.
The stark reality is that given the backlog of reports, some critical, of children at risk that will have built up over the Christmas/New Year holidays, if you make contact with the NSPCC later today it is unlikely that any enquiries will be made by your dm's local SS until sometime next week at the earliest.
I would therefore suggest that you spend today concentrating on your own health and make it your prority to go see a doctor as you've planned.
When you see your dm on Tuesday, sit down with her, outline your concerns about your dsis, be honest with her about what you know has been taking place, and urge her to make contact with SS as a matter of urgency.
Your dm needs to know that she has nothing to fear from asking for help with, and for, your dsis who is exhibiting behaviour that is not uncommon in young people who have suffered historic sexual abuse*.
From what you've said, your dm is doing everything within her power to keep her family together but she needs the benefit of SS intervention to enable her to promote your dsis's welfare and wellbeing.
It's probable that past sexual abuse is causing your dsis to act out but this extreme attention seeking behaviour may be exascerbated by the amount of time your dm is required to attend to the needs of your disabled db an younger siblings.
I cannot emphasise enough that voluntarily contacting SS is NOT going to put your dm at risk of losing her dc, but she should give thought to how it will appear to others if, for example, your dsis is found to be pg at 14 or 15.
Should this occur, it's probable it will emerge that your dsis has been having underage sex with a 23yo man while under your dm's roof and IMO your dm has more than enough to contend with without having that visited on her.
If, after hearing your full and frank disclosure, your dm is unwilling to make contact with SS or your dsis's headteacher, report your concerns to the NSPCC and they will subsequently notify your dm's local authority who are duty bound to investigate any reports passed to them by that agency.
However, please be aware that if your dm and your dsis deny that anything untoward is taking place or refuse any offer of help, and if your dsis's school have no concerns about her behaviour, that will most probably be the end of the matter and you will have to content yourself with knowing that you've done everything possible to demonstrate your love and concern for your dm and dsis.