Note - this is from the UK perspective, and written as what I roughly regard as their 'normal' behaviour. Sources are first, second and third hand.
1/ The 'warning signals'.
Most of Social's infomation comes from other agencies - health, education and police. As mentioned before, some families can be very good at hiding / coping with these. If the kids are old / mature enough to take up the slack, money isn't an issue and they don't give any obvious reasons to be noticed, it could go on for months or even years.
Schools look for things like dirty clothes, poor diet, lack of parental support (letters not signed etc.) Stuff like truancy and poor behaviour could also cause the school to call in the parent, then to learn the parent's cracking up.
Health services are the most obvious choice. One of the parent's contacts (GP etc.) sees they're falling apart, and decides to call in the Social. Also, it could be on of the parent's friends / relatives who talks to their GP about their worries in confidence, and the doctor cites 'child protection' as an excuse to break it.
Police often find out, usually in situations where they're investigating other crimes. But unless one of the kids has taken to theiving or whatnot, it's unlikely in your desired situation.
2/ Investigation.
From what I've seen, they start out with the initial meeting with the parent at the school / surgery / office. This will be a strict issue-based meet ('Johnny isn't turning up for school etc.'). They won't automatically think the parent is 'unfit', but if they don't 'follow the script' they might start to blame the parent.
If suspicions are roused, they'll contact the Social. I suspect they start by hitting the records - has the family crossed their path before? I have seen from personal experience where Social have taken a kid away on not much more evidence than the parent had been in care a decade before and a couple of little things taken out of context.
They'll also hit the families' GP and kid's schools for infomation. They may call in one of the kids 'to chat' with a Social Worker while there - but they do not say they are from the Social.
If this turns up enough 'dirt', here comes the Home Visit (capitals intentional). They're sneaky. Will bend over backwards to get this. Make meeting at the parent's home the best thing in the whole world. For they want the Home Visit.
Not only does it put the parent at ease (as it's their territory), but gives them a great chance to spy. If given half a chance, they'll go in the kitchen cupboards (have you got food?) ask to use the loo (hopefully upstairs, so they can look at the bedrooms) etc. Again, unless pressed, they won't admit to being from the Social - will usually come with a 'trusted professional' (GP, teacher etc.) and simply be introduced as 'a collegue'.
3/ Showing their colours.
If they've found enough dirt, Social goes in for the attack. Their two tactics are fraud and force.
Fraud - they see the parent a few more times. Build up a rapport. Suggest they 'need help with a few things'. Suss out any relatives that live locally and try to get them on Social's side. If they haven't done so already, they'll interview the kids and other 'significant adults' alone. Only once they've got their foot in the door will they admit to being Social.
Force - Attack of letters and phone calls to parent. Random home visits by Social Workers who are almost as insistant as Bailiffs in coming inside. Missed meetings? Oh dear, Social don't like getting ignored. If they think the kids are 'in danger', they might progress with applying for stuff like Care Orders.
If you want better details for your WIP, feel free to PM me. There's stuff that while I don't want it plastered over an open forum, don't mind giving privately. You're actually doing research into it, which is better than 80% of people who write on the subject!