As a working single parent, you get 30 hours free childcare from 2, not 15 hours (term time only).
I would say that if possible, you would all be better served to support your daughter to stay in her private rental (financially if needs be) whilst she waits for social housing, rather than move her in with you.
Once the new baby is 1, and she can more easily go back to work, she’ll find that between free hours, the single persons/ children and childcare elements, child benefit of universal credit and the UC taper down for wages, it should be very manageable.
The difficulty is that UC for childcare is paid in arrears and universal credit assessment window doesn’t always match billing periods, so you may find she needs a little bit of short term help (financially) for the first month or two on return to work whilst things settle into routine.
All of the above from experience. I’m a single mum, unexpected separation when DS was 6 months, went back to work when he was 1 year. it’s hard, but DS is 7 now and we’re doing okay!