The filthiest homes I've ever been to are the homes of nurses.
I realise that "not all nurses..."
This is just my own observation.
I'm not talking about untidiness. I'm talking dirt, stench, mould, fridges that are hazmat scenes, rugs of indeterminate colour, dogs in dire need of grooming, gardens full of dog poo, every surface piled high with goodness knows what, sinks full of dishes, bins overflowing.
Maybe there's something about the stressful working environment that makes it hard to keep all the plates spinning at home. I'd say the fact the husband basically works away has a lot to do with this woman's mental state. She's not coping well at all, basically on her own, neither single nor partnered really.
The worst home I've ever been in (friend was a nurse) was one where this was the setup. Husband away for work (in the oil industry), friend working PT and also trying to manage the home. In the case of my friend, the husband had a girlfriend for many years, someone who would fly to see him wherever he was working.
OP, I'd ask her to make time for a chat, once this holiday is over. I'd open with a casual observation that life with children is exhausting, and it's so hard when you're basically trying to hold it all together on your own. See where it goes from there. I'd push for her to hire a cleaner after an initial big sort out. Repeat that your suggestions come from a place of love and no judgement whatsoever.
Sadly, it's been my experience that medical professionals are able to get away with all sorts of neglectful treatment of children - some social workers seem to have rather too much respect for the profession and don't push too hard.
However, if your friend is really resistant to your suggestions, I'd call the children's school and ask to speak to the safeguarding lead. I'd tell that person that you are aware that this woman and her children are often well turned out and that this is stupefyingly miraculous, given the horrific state of the house, which you should describe in as much detail as possible. If you have any photos to show them, that would be ideal, but perhaps that ship has sailed.
The fact that the family managed to turn up well dressed and well presented shows that she is aware to some extent of what the reality is, but she's trying to present a facade to the outside world. I think she knows she needs help. Maybe her invitation to you was a well disguised cry for help?