Nanny, you asked about the sleep deprived mother. It would all depend on; how old baby is, how much support there is at home, what the mother wants to do.
Formula isn't a magic sleep inducing substance so I would caution that the mum might find it won't suddenly make a baby sleep through.
If it's a newborn of say 6 weeks I'd explain about growth spurts and constant feeding to increase supply.
If mum wants to introduce a bottle I'd say it that yes some mums do this but it might affect supply. Obviously I can't say one wAy or another if it will or won't. It's not my decision to make.
I'd explain about cluster feeding if a mum is worried about spending hours in evening feeding. I tell her it won't last forever.
I tell her we meet every week to support one another in a safe environment where all aspects of BF can be discussed. Because sometimes just being with others in same boat helps.
For a mum with an older baby or toddler not sleeping well I'd suggest the health visitor, because my training is limited to breastfeeding. BF babies can sleep through the night from quite young and its really annoying having to fight the misconception that just because a BF baby feeds day and night in the first weeks of life they will forever feed in this pattern until you introduce formula.
Mostly we are their to listen, offer information and signpost to either GP, midwife, HV etc.