It sounds like everyone involved here is really struggling and is in need of some compassion and help...
Your DD, in her early 20s, with a brain injury, quite possibly with PND, and without a relationship which sounds to have been quite supportive for her.
Her ex (baby's dad), in his early 20s, probably with a lot of growing up still to do, trying to keep employed in a difficult financial climate, while adjusting to parenthood after the end of a relationship.
You, trying to support everyone while also keeping your own work etc going.
And your grandson, who's stuck in the middle of this situation and will desperately need some stability and love.
It may be that your daughter genuinely cannot cope. Brain injuries can have lasting effects which can be masked or be less evident until something comes along to expose them. Being a parent requires a lot of work by a brain... Balancing schedules for feeding / sleeping / changing nappies, keeping the environment safe, making sure there's enough nappies / clean clothes / formula / etc - that's a lot of executive functioning and switching between tasks. Then there's detecting, interpreting and responding to a baby's cues for hunger / distress / tiredness / interactions. Then there's coping with the fatigue, compressed opportunities for self-care (like having a shower or something to eat)... Brain injuries can also increase the risk of depression, anxiety, emotional lability, etc.
It sounds like your grandson's dad is trying, so I don't think it's right for people to be too hard on him. Being a single parent is hard, especially a young single parent, and especially if his own family isn't very supportive. If he's trying to hold down a job, earn money to move out from his parents', and be primary caregiver for almost half the week while maintaining positive relations with his ex-partner's family, he's already doing quite a bit more than a lot of young men would be doing in his situation. Could he do more? Well, yes, but that doesn't mean he isn't trying. With some additional support (eg peer support), he might feel more able to do more.
I think it's important to get all of you some support. Health visitors, GPs and social work should have access to relevant resources, including specialist services and third sector organisations. If your daughter isn't already, she might benefit from some contact with a brain injury charity as well as NHS community brain injury services. Your grandson's dad might benefit from getting to know other young dads through parent and baby groups, for example, where he can learn more about juggling the competing demands and make some friends who can support him and encourage him.
If, despite everyone's best efforts, your daughter still cannot cope with being responsible for her child, then you and your grandson's dad may need to consider how to find a solution that provides safety and stability for your grandson. How do you feel about his suggestion of shared custody? Would you prefer for him to be the resident parent while you provide childcare? Would you prefer something else? If the situation gets to this stage, you might need to seek legal advice about your options. I don't think there's a 'right answer' here, just what works for you all.