You say you cannot win, but there are no winners here.
Either your DP has to see his nephew go into care, something which I would personally find unthinkable. Or he has to not live with his child. That is a terrible choice to put on him. You have a similarly awful choice.
So try to limit that mindset and try to tackle the problem together, rather than this "you win, I win" attitude.
When will baby be born? Can you renovate the loft faster than the extension? If you were planning on doing both you could do that first so nephew has his own space as priority.
Could DP and nephew move into grandparent's house temporarily until things settle a bit?
You mention finances aren't a problem, so could DH sort a 2 bed flat nearby for him and nephew and test the waters?
It might be too much and you might have to separate, but I would try any and all solutions, even if not ideal, before splitting up.
*Edit, just seen your update about DH lying to social services. Yeah, that is foolish and my sympathy for him has somewhat lessened. You cannot just hide your wife, 2 step children and a newborn baby? That is a recipe for disaster.