It's tempting to think that there may be a way to use this incident as a turning point whereby you are suddenly happy families again, brother in work, mum free to live her life etc, but realistically it's not going to happen.
While your brother's apology may be sincere, it's unlikely that he'll feel remorseful for very long and he'll carry on using your mum as his personal drudge for as long as she's willing to tolerate it.
The only person who can change this situation is your mum and, as cruel as it may seem, to some extent you have got to withdraw from her because, in highlighting your brother's shortcomings, you're providing a distraction and standing in the way of her seeing that she currently has a needlessly impoverished quality of life.
When chatting with your mum keep everything upbeat. Don't mention the war your brother. Don't ask about him. If she says anything negative about him or her situation, make a polite non-committal response (along the lines of 'o that's interesting') and change the subject.
With summer coming up, could you include your mum on a couple of all-day outings with no specified return time? Maybe a car trip to the coast, a meandering drive home with plenty of nice scenery to look at, and maybe a stop off at a nice country pub or an easily rectifiable breakdown of an hour or so thus ensuring that she isn't back to cook dinner/hand over her car etc. Maybe even a short break somewhere that has unreliable mobile phone contact?
The objective is to get her into a space where she is enjoying her unfettered state in the hope that she'll realise what's she's missing - and why she's missing it -with no nudging or prodding from you.
I'm well aware that it's easier said than done; you'll need to be forebearing and tolerant with no guarantee that your patience will be rewarded, but at least you won't be tearing yourself to bits over your brother and adding to your mum's problems.
My plan B would be to stick brother's details on a dating for the desperate site in the hope that there'll be a sucker taker that might, even temporarily, cause him to get his act together and take some heat off your mum, and get mum engaged in as many local activities as possible in the hope she'll be swept off her feet by a charmer who'll kick your brother into touch.
We can but dream, but real life can be stranger than fiction and miracles do happen.