It would depend if she has legal custody and whether they were conditions on her having it, i.e. appropriate accommodation for the children.
If he thinks the situation is now inappropriate, he would have to challenge the custody on these grounds (or apply for custody if it is an informal arrangement at present). The law may not think it is inappropriate, you would have to find out the legalities, as many people use a downstairs room as a separate bedroom (at uni people often converted the lounge into two bedrooms with a curtain!).
I don't see that he could force her to try and re-get a bigger council house (who's list she will be further down now) unless she sees it as the only way he won't try and take the children.
The Citizens Advice Bureau (
http://www.citizensadvice.org.uk/) provide free and confidential advice, so they could advise of all the avenues open to you and your partner. They would also know the exact laws regarding children and overcrowding.
Good luck