Why SS? An investigation just puts a struggling Mum under greater stress without actually providing her with any support.
Why not the HV service? They have quick access to genuine support for struggling parents that can be accessed fast - such as parenting groups, courses, early (2 years +) free nursery places to give a parent a bit of a break etc, etc. The HV service can be accessed via the GP. They often send in a homestart volunteer - someone with parenting experience to offer real support over a period of months to someone feeling lonely, isolated, judged, and overwhelmed by the lone parent experience.
Help not threats is what's needed in situations like this, at least in the first instance. I've lost count of the number of Mum's I've spoken to IRL who have been at their wits end cos a kid's behavior was caused by special needs yet "the system" wasn't supporting them.
With a 2.5 year old - a HV is much better placed than a SW to check that parental stress isn't down to undiagnosed special needs in the child or post natal depression/malnutrition in the mother. (not eating properly cos benefits have messed up is all too common for single parents!). SW's don't have the same skill set. If it is abysmal parenting without mitigation, then trust me, any HV worth their SALT will call in SS, and be taken much more seriously than a member of the public in making sure that child is made safe asap.
Smacking is legal - you don't have to like it, but it is. Swearing at your kid as a one off when at the end of your tether, also isn't illegal - morally I think it's repugnant, but I can understand how someone without money, emotional support, a stable home etc might be driven to it just the once.
A SS investigation in and of itself, causes it's own VERY significant damage to a family unit, from which this Mum & her child may never recover. Contrary to popular belief it doesn't automatically bring about this amazing utopian upbringing for children that everyone assumes it does (look at the number of kids in care who wind up trafficked each year, or the educational attainment levels of looked after children to see what I mean). SWs have no influence over health services - which is what he Mum may need if she is at the end of her tether.
I honestly think that if you think a child is at immediate physical risk of harm then the police are the best agency to call. If you see a beating (as opposed to the legal smacked bum or tap on the hand) dial 999, don't mess about ffs.