I hate to say this, OP, but often we go in patterns. You ended up married to a controlling man who abuses you in a lot of ways (even if it’s not physically, which unfortunately, is all some older generations, like maybe your mother, recognize). You seem to have people-pleasing tendencies - he treats you like absolute shite, then you worry if not responding to a text for six hours is too mean? First of all, it definitely isn’t. Believe me, if he gets partial custody, I’m sure he’ll ignore your messages for hours or days, even when he has the kids. And I don’t say that to be cruel, re: people-pleasing, because this is engrained in you and it will take a while to break out of the cycle. And it sounds like maybe this comes from your mother. She seems to be a people-pleaser, too, married to a man she’s either been beaten down by or just can’t stand up to (I think many of us here, if our husband told us someday that our abused adult DC and DGC couldn’t stay with us, would tell him he’s completely in the wrong and to fuck off, not just bow to his wishes). She wanted to please you so told you that you could come stay, she wants to please your step-father so now she’s trying to get rid of you without thinking of your safety, and it seems like never once is she thinking about what’s actually RIGHT in this situation.
People-pleasing is dangerous. It gets women hurt and killed every day. It keeps them in abusive relationships.
Also, we read here every day from women who say, “Oh I can’t possibly say too much to my real-life friends. They have so much going on themselves.” If my friend was trying to leave an abusive relationship, I would want to know everything ESPECIALLY if she was facing homelessness. Let them make their own decisions about how much help they can give; it’s not fair or right for you to decide for them because you’re trying to, well, be a people-pleaser in advance.
But yes, you must make it clear to your mother that no matter what, you are not going back to that house. You may jeopardize your UC claim (I'm not 100% sure of this, I admit), and if domestic violence charities are involved enough that they recommended you take your eldest out of school in order to escape your household, they may determine that returning to your home would be endangering your children. I’m sorry and I’m not trying to scare you; I unfortunately know someone who had this happen, where she tried to leave her abusive husband, made the decision to return when we all begged her not to (no one was angry as I know the average woman tries several times to leave), but then she had an SS case opened against her (when HE was the arsehole, it was all so fucking awful). It was just mad.
Hopefully you get the second house and this is moot, but if not, perhaps consider the temporary accommodation and in two weeks, if you are then unintentionally homeless, then you will jump to the top of the priority list. And please, do NOT let people put pressure on you about how this could stress the children. In the end, this will be less than a month of their lives, versus if they’d had to stay for years and grow up in a terrible household where their dad treated their mum like shite and it would have affected all their adult relationships.