Which is correct and why?
Them looking after the kids was a great help.
Them looking after the kids were a great help.
Them looking after the kids was a great help = Their looking after the kids was a great help, is correct, I think, non-standard though. 
Neither one is correct.
It should be, “Their looking after the kids was a great help.”
The subject is a gerund, so it takes the third person singular form of the verb.
I’m not sure I understand why they are both incorrect. “Their”, there, sounds awfully formal. What would be the informal equivalent?
Is the use of “them” incorrect here?
"We didn’t like them coming in the police station, they used to come into our parade room where our men were having food, and you can imagine them coming in a bit lousy and so on. "
Should it be “We didn’t like their coming in the police station”?
Do you mean ‘looking at the kids’ or ‘looking after the kids’?[YSaerTTEW443543]
TOEIC listening, photographs: A designer loft[YSaerTTEW443543]
“Their” is not “awfully formal” in that context. “Them” just sounds awfully crass.
“Their” can be used in these sentences also, but “them” is also fine.
The difference is that in your original sentences “them looking” is in the subject position, and so the pronoun should not receive object marking.
In your later examples, “them” is in object position, and so it can be marked as an object.
Right. It was late at night. It should be “after”.
Which context are you imagining?
Why am I suddenly imagining something? You started out by giving a couple of sentences containing the word “them” where “their” should be used. Those sentences are the context. You’re acting as if you’d never posted the sentences and that I was hallucinating.
Sorry, but do you mean “in the context of talking about childminders…”?
Could we say “The act/action/favour/ of them looking after the kids was a great help.” If so, why?
I know that the object case is often used before the gerund in the middle of a sentence (We apprecate you coming so quickly). In fact, it’s used so often that some people do consider to “overly formal” these days to use the possessive. However, at the beginning of a sentence like that, I’d go with the possessive without hesitation.