My brother's house was like this when he had kids - the kids are all fully grown now - 3 kids, wife was a brilliant artist, he worked long hours in the city. Both parents spent a lot of time with the children, talking to them, taking them away on holidays, spending time together at home, etc. My brother was a really good cook but didn't see the need to wash up, they were both very funny and had people over all the timeThe kids had good clothes. So it all sounds quite similar. It hacked me off but they were absolutely set in their ways, and the kids were amazing as kids and as adults are all following their passions with their careers, lots of friends, good social lives (not sure about state of their houses!). I think that it is probably true both that the children were happy with everything in their life other than the state of their house - they would have preferred a tidier house - but that any SS involvement would have not helped at all on any level.
Child development research indicates that the one key thing necessary for children to thrive is love/connection - having someone who knows you, loves you, understands you, talks to you, supports you with problems, supports you with your passions etc. The second thing is a mix of high expectations of your child in terms of behaviour and school work with high levels of support. I have not seen research about the link between messy houses in childhood and bad mental health in adults - and perhaps one of the posters who say that there is lots of evidence could link some (such as @Janedoe82 - can you link actual research as opposed to articles?). I think this is because with mess and filth at home, there are going to be variables - if the parents are otherwise engaged with the children and meeting emotional and physical needs, this is a far cry from not cleaning because the parent has MH problems or high or drunk. Also, mess affects some personality types more than others.
The other thing to note here is that doctors are scientists - and SWs are not scientists - doctors deal with illness and life and death and they are going to be aware of risks, and they are also more likely to be aware of up to date child development research. I realise that the SWs on the thread will be clutching their pearls as they read this, but I think it to be true. Child protection policies in the UK are not in accordance with up-to-date child development research - some of the policies haven't changed since the 1970s.
So going back to the risks in this house - as long as mould isn't served up to dc, and plates are washed before putting food on them (as happened at my brother's) then the physical impact from a scientific and rational point of view is low, however grim it may seems to most of us. Same for the comments about toxoplasmosis - if your toddler has ever played on a beach or in mud or in the park or your garden, chances are they will have been exposed to serious levels of germs including toxo and including super bugs. Unless you hover over them all the time and try to wash the sand or mud before they play, they are being exposed to serious germs. Children in house where there are overflowing litter trays will be aware to stay away from them.
I am not defending messy or filthy homes at all, incidentally. In an ideal world there would be the love and connection and the lovely clean home too.
Someone said upthread that if SS were involved, the children would not be removed. In fact, this would depend on how things went, on the individual social worker and their manager. The threat of removal would loom large however and would cause high levels of stress - stress between parents is not good for children either and can cause crises and all this could lead to long term MH problems just as mess could.
And if you are concerned about MH of children, we know that screen time, iphones, violent and oversexualised video games and books, unfettered access to social media are utterly, utterly disastrous for children. Yet it continues and i would bet that half the posters on here shrieking "report" will allow their dc the above so that they "fit in" with other kids. I think dealing with the above issues and other horrendous things facing our kids re grooming, drugs, gangs, are more important than mould on dishes overflowing litter trays - even though those things are not good either - if I were the OP I would suggest to the friend friend that she should get a cleaner.