I might get slated for this, but I can't see what's really so awful about the mum in the OP?
I wouldn't see any of the three "offensive" things mentioned as all that judgement-worthy in and of themselves. I guess for Mummikins to be disturbed by it, the context must have been bad, but just on the facts in the OP it doesn't sound awful to me.
So she was on facebook in the library and not paying attention to her son. But was he looking for her attention just then? Or might he have been drinking his bottle and / or playing with buggy toys while she had some down time? Maybe they'd just come from the park. Maybe she'd spent all day playing and cuddling with him. Everyone here spends time on the computer too, would it have been different if she'd been on MN in her living room rather than FB in the library?
She was swearing in front of him. Not to everyone's taste of course, but does that necessarily mean she's an awful mom? Was she swearing at him or about him, or just using swear words in conversation out of habit like a lot of people do? Was she saying "ct" and "motherfer" or "damn" and "hell"?
And the "little tramp" thing, well, others have already said that they've said the same or similar to their kids in a playful, affectionate way. Could that not have been the case here?
OK, if he was crying for her attention and she was ignoring him and telling her friend what a f*er his dad is and then he dropped the bottle and she screamed at him that he was a tramp, then yes, she's a terrible mother and the OP is right to judge.
But if that had been the case I'm guessing those details would have made it into the story.