Hi OP,
Your daughter and grandchild are very lucky to have your help and support but you do need to find a way forward to make sure that the child/mum relationship works so that little one can grow up happy and feeling loved and secure. You mention your concerns about your daughter experiencing regrets later. However as an outsider I would be more concerned about how the child will feel about their relationship with their mum.
That being said you can't force your daughter to have a good relationship with her child you can only nurture it. Your daughter may not be mature enough to be a good mum 24/7 and you are (wonderful and) willing to take up the slack. However neglecting her responsibilities entirely is not an option either. I think you and your daughter need to sit down together and form an agreement of when she will be responsible. If you ask her to help you work out a schedule of how she will balance her social life/college work and her child she will hopefully come up with reasonable suggestions. Perhaps three days in the week going out with her friends after college and 4 days being at home with her child and splitting each weekend day in half so she is either with her child all day and able to go out in the evening or out all morning and comes home in the afternoon to take over? Ask her what she thinks is reasonable time to spend with her child, bearing in mind she is at college and see what she comes up with. Encourage her to be realistic as a plan that she can't stick to could knock her confidence further.
For the child's well being I think it is vital that although the time with their mum may be limited that she has a warm and positive relationship with her. You can help by always being pleased to see her "here's mummy!" and "we've missed you mummy..." come and see what x wanted to show you" etc
I'm sure you are a wonderful mum to your daughter and that you support her relationship with her child well but it can be tricky to navigate this and not undermine your daughter as mum when you all live in the same house and you have the benefit of maturity and experience. It is important for your daughters confidence that she is "mum" and that you defer to her in decisions "we'll have to ask mummy what she thinks about that..../if you are allowed...../if you can have...."
You can boost your daughters confidence sometimes when you are all together by suggesting in the park for example "lifting you up - that's a job for mummy - nanny's exhausted!" or "mummy's the best at that" etc. You need to remind her what she is good at "mum has the best cuddles", "mum can always make you laugh.."
If you are kindly having the child overnight and during the day when your daughter goes to college I wouldn't worry too much about time together in the mornings/greetings etc as if your daughter is in a rush and not available really then that will probably be more disruptive for little one than low key and slipping out unnoticed.
Apologies if I am wrong but your post sounds a little like you would allow your daughter to opt out of being a mum and maybe take over the role? I do apologise if this sounds like a hurtful or spiteful comment that is not how it is meant but you have to be absolutely 100% clear to yourself and your daughter that you are not this child's mother. Your daughter is mum and this child needs her. You will be a wonderful loving and caring nan (and of course mum to your daughter)- but little one needs a relationship with their mum - unless the relationship is so toxic and negative that it will be more damaging to the child to maintain the relationship. If you seem to be offering to step-in as mum (even though not in name) then your daughter may consider that you would be a "better" one and leave you to it. She is mum and it is her job to be the best one she can be and yours as a caring nanna to help her to do that. Helping your daughter to be a good mum will be doing her far more favours than doing it for her.
I have worked in family support including with young mums who need support to manage, find their confidence and a positive relationship with their child and I think you can nurture the parent/child bond. A supportive nan and family can help this relationship to succeed where it wouldn't without support.