You don't get 70% of childcare paid just because you're a lone parent. It's still salary dependent. As soon as you start earning more than NMW the percentage is reduced.
I managed to put two through full-time childcare on an income of about £32,000. It was desperately hard at times and has left me hungry and without the means to pay for necessary dental care and replacement glasses. But I managed. With one child, your friend should manage a little better. IT won't be easy, particularly in London, but it should be do-able, especially if she can come to an arrangement with another parent or if she can get some maintenance.
She should definitely go for maintenance, since the child has a right to it. She doesn't need to name the father on the BC for that. If she chooses not to name him he can apply for PR via court and would almost certainly be granted it; the question is whether he would bother or not. If he's a lazy arse or a controlling type it may be best to leave him off since the extra steps he would need to take would give her time to counter his moves. Why is he not around? If he doesn't want to be, don't name him and let him apply for PR if he can be bothered. If it's because he doesn't know about the baby though, she really ought to tell him. He should not be denied the chance to know his child since it's actually about the child's right to know both parents rather than his rights over the child IYSWIM. He may actually step up to the mark.
Lastly, just because children whose parents are separated have a greater likelihood of less successful outcomes in life, does NOT mean that this is a sure thing. THe biggest effect comes when the parents separate. Understandably there is a lot of change and often a lot of bad behaviour at this time. This is what does the damage, NOT the parents being split up. Things usually settle down in time. The fact that the relationship is over before the child is even born means all this has been circumvented. Once things are stable, if you factor in income and education, the differences between children whose parents are still together and children from 'broken homes' (stupid phrase) are negligible.
Finally, the relationship with the NRP is open to debate. It's not quite as simple as contact is good, non-contact is bad. Recent research shows that sporadic unreliable contact is as damaging as no contact at all. And children who are brought up in otherwise loving households with healthy relationships with several adults (e.g. extended family) do NOT usually suffer from behavioural problems or low self-esteem if there is no father around. It is a lot more complex than that. Children from families in which there is no father figure are statistically more likely to be living in poverty and social deprivation, which skews the figures.
Rejection is the main issue, which IMO is why poor contact is worse than none at all as it is a continued form of rejection over and over again. Where a lot of parents with crappy XPs go wrong however, is trying to eradicate the NRP from the picture even when contact is non-existent. A child has a natural curiosity about his or her parentage and this curiosity should be satisfied. It is important to find positive and neutral qualities and characteristics to latch on to so the child doesn't feel ashamed or apologetic for half its inheritance. This isn't always easy but it does not mean lying or covering up for bad behaviour. Little things like "your dad was always good at maths" or "yes, your dad's favourite crisps were salt and vinegar too" can make all the difference and help give a child a sense of identity.
I wish your friend all the best.