I also find her fairly irritating with the little girl voice and endless terrible advertisements for stuff like kitchen roll and Bisto gravy and terrible chickpea curries made with sauce out of a jar (and a handwritten menu, for goodness’ sake), but… I am going to offer a slightly more charitable perspective on the separation/anxiety thing.
Izzy comes from a family of very talented classical musicians (her parents ran/run? a music school) and I think is the youngest daughter with three older brothers. Guy is a professional cellist, Magnus a professional violinist, and Izzy was performing as a professional violinist when she met Harry.
Her three brothers were all boy choristers at Kings Cambridge (ie, the one that does the Christmas Day service on tv each year). Her brother Rupert was an unbelievably gifted horn player who was studying at the Guildhall and no doubt was also destined for an impressive career as a professional musician. I think he was quite a character - very witty and full of life. Whilst he was at the Guildhall he had a very serious car accident (I think in the middle of the night) when Izzy was in her teens which caused serious, traumatic and long-lasting brain injuries. The impact of this on him and on his family must have been profound and it is not entirely surprising that Izzy has struggled with anxiety, particularly separation anxiety in the years which have followed.
I don’t disagree that she could probably benefit from some therapeutic support or that her own mental health issues might not be having an impact on her children (she does seem to spend a lot of time projecting anxieties onto Lola and Kit) but it’s perhaps not completely surprising given this deeply traumatic event in her family history which would have been a lot for anyone to process at any time in their lives, but especially in their formative teenage years. Their family has not only had to come to terms with the trauma of the accident (the suddenness, the timing etc), but also have had to process the loss of the son and brother they knew and the life he was going to lead. By all accounts he still seems to be quite the character - fun and full of life - and he does still seem to get great enjoyment from music but Izzy has posted on a number of occasions how the accident has changed him and the grief and sadness that comes from that. I think it must be a bit like a lifelong bereavement in a way (which is not to say that they cannot also be thankful everyday that he survived the accident).
Anyway, she is fairly annoying and the permissive and mindful parenting doesn’t seem to be working out too well if her kids are drawing all over the furniture (I’m not sure I would tolerate that although I wonder whether it is a coincidence that is has happened whilst Harry appears to be away from home for a prolonged period of time - Harry seems to have a close bond with Kit and Izzy with Lola), but I offer the above as a kinder view of why Izzy may have some of the mental health struggles she does. You can find someone massively irritating and still be kind.
really puts me off the Judds
I’d have thought with all her mindfulness she preaches about she’d be mindful of her own child’s privacy, not just hopping on the instamum bandwagon of posting intimate details of your child’s life 
