Agree with SpiceMonster.
Re. your point TM, if the parent whom the children live with gets a new partner, and he or she chooses to allow the new partner to move in, the children are forced to recognise the new partner's role in their lives, whether as step-parent or just someone they live with. They have no choice because the parent they live with has taken this decision on their behalf whether they like it or not.
But when the NRP has a new partner, it does not necessarily follow that the children have to meet this person straight away or indeed spend any time with them. Because they don't live with the new partner of the NRP, there is no specific need for them to meet them at all, indefinitely. Of course, most people would wish their children to meet their new partner at some point and to grow to accept them, but that is a different issue.
This may seem inequal between the Parent with Residence and the NRP, and it is often the father who is the NRP, but it's not about equality for the parents, it's about what's best for the children first, and then everyone else concerned. If children lived with their father and the mother got a new boyfriend, the father would feel wary about his children meeting him.
Whatever the background, regardless of whose decision it was to part, when you have had a relationship breakup, it is not uncommon to be a bit suspicious about people playing happy families (maybe this is wrong but you can see why people have mistrust).
Someone in Charlotte's situation with children who are only little, one is a baby, and an ex who does not sound very responsible, are bound to feel very protective of their children.
As it turns out, none of this is relevant to the OP. Charlotte is very generous towards and tolerant of her ex and his new partner. She just wishes that she could have some privacy to speak to him alone about their children instead of having to always speak to him in front of his new partner who has no formal role or responsibility in her children's lives. The fact that she is always present suggests that his new partner has a problem of some kind.
Under any circumstances, it is irritating to have to treat two people as one. Such inseparable 'TomKat' couples should get a life (or rather some kind of separate lives).
To me, it seems that Charlotte's ex and his new partner are ganging up on her, two against one. To criticise her for not being dressed, when the new partner arrived uninvited, is outrageous.