I can't work out what you're saying the arrangement is.
No, I personally don't like "nesting" which I think is the popular term for children staying in one place and the parents moving in and out of it.
I don't think it's unfair to children though - I think it's a lovely idea - if you can afford 3 houses! No way would I share the other house (for the parent not with kids that week) with my ex!
But that's not what your partner does.
He visits them in their home, doesn't live with them.
I thought you said it was unfair for the children to have two homes? And that's what I said I don't think is unfair at all. Many children have this and accept it happily.
I think he has been foolish (if understandably so) about not sorting this out properly before the boyfriend came on the scene.
Now I could pop in to my ex's house - the old marital home, for which I have a charge on the equity, as I generously allowed him not to have to sell up, for continuity for my daughter. I don't, but emotionally I could. But... would I want to if her new gf was there, offering me a cup of tea? Maybe. But maybe not. And I haven't already expressed upset at the new partner, or think its an OW/OM. Is he really going to want to read to the kids when this guy is walking around in his boxers and slipping his arm round the wife's waist?
And is the new bf going to say "why the fuck do we have your ex wandering in and out, honey?"
Get it sorted legally. Protect his share of the house. Then stop paying the mortgage. Put it towards renting a 2 bed, and get 50/50 shared care agreed.
If she can't pay the mortgage, then sell up, or she can get her bf to contribute.
He is a fool if he doesn't sort this out properly.