The children from programmes from that era are old enough to have their own social media and some have uploaded their own views about it. Most that I have seen tend to be "Lol look how naughty I was, this is funny, look at me now!" (They seem to very rarely be doing anything of note now) - but I did see a video of someone who had been on an episode of Hoarders as a teenager, and felt that the entire process had been exploitative, disruptive and did not help long term.
Of course if someone doesn't want their name linked to this old TV programme they are unlikely to post openly, so I wouldn't take that as a given.
Some episodes have been removed from re-airing due to privacy concerns, which may have been at the children's request, and the families' names removed from the Supernanny wiki.
As far as I can tell from what I've seen from people sharing their own experience plus a few comments on the wiki, some families found that the approach did help calm things down somewhat, and combined with the children growing up there was a longer term positive effect, but many did fall back into some state of chaos. Some families sold stories to newspapers/other media in the following years claiming that the producers wound up the children or staged some of the incidents shown. It's hard to tell whether this is genuine gripe because they were misled or a skewed/unreliable narrative of what actually happened. Some of the children ended up committing crimes and going to prison. Several families went on to have more children. Many of the couples divorced/separated. Many of the children on the programmes who have posted about their experience state that they do not have much if any contact with their parents as adults.
Interestingly this was also a topic on the podcast interview, and Jo said that she didn't feel responsible for whether or not parents fall back into bad habits after she leaves, which I thought was a bit simplistic - I felt she was implying if the parents had stuck to her advice perfectly and used it always, then every problem they have ever had will be solved, and that seems unlikely. It also ignores the fact that it's not completely random that a family ends up in the crisis point they are usually at at the start of the programme. Most of Jo's advice is extremely basic and the first parenting advice you'll find in any book, article or course - clear boundaries, connection, structure and positive role models. So if you're not managing those things often it's because of underlying issues e.g. own childhood trauma, lack of own role models, extreme stress, emotional difficulties, very poor self-awareness, MH problems/ND etc. You can override things like that for a very short intense sprint like a 3-week TV programme with an expert coaching you all the time, but parenting is a marathon not a sprint. And as others pointed out, TV editing is such that you can easily show the worst parts of the "before" week and the best moments of the "after" week and make it look like Jo's methods have totally transformed the children but that is unrealistic in such a short space of time. And as you usually see during week 2, it only takes a small amount of slack in the plan for things to unravel, and parents are only human and probably humans with issues or they wouldn't have got there in the first place.