It's difficult to advise without knowing much about the type of affair it was or the particular circumstances. There are however two huge additional negatives to this story. If a man has been unfaithful this early on in marriage, it suggests he is not ready to be monogamous and that the motivation for infidelity is more to do with not being ready to commit to one woman.
The second negative (and it is the biggest one) is that if someone has seen the pain a first discovered infidelity causes - and goes on to do it again - it suggests that they really cannot love enough.
With all those circumstances, it is better for your sister to cut her losses and acknowledge that this man is unlikely to be faithful - to her or anybody.
This last point is crucial. Sadly, betrayed spouses tend to beat themselves up for lacking in some way and causing the infidelity. It is essential that you drive the message home that this is in no way your sister's fault.
In her particular case, her H is probably not ready to be faithful to anybody.
The other point that might be crucial to understand as her helpers, is that in many cases, there were no pre-existing problems in the marriage before an affair happened. People on the outside often assume that there must have been, but in many cases it is just all about the betraying spouse's response to an opportunity. However content they are with their home situation, they just cannot pass up the opportunity of an adventure, or the lure of someone new.
If there were gremlins in the marriage, all parties need to distinguish between their shared responsibility for those problems and the response to the difficulties. This is really important. Infidelity is a spectacularly useless way of dealing with grievances in a marriage, whereas airing those issues and trying to resolve them as a team is the adult response.
If this has never happened before to you or your Mum - and you are trying to help her, the very best you can do is to listen and try not to judge her. If it's a recent discovery, she will be feeling shock, a lack of safety, some idiocy for forgiving before, anger, sadness and possibly some denial.
Mother her, offer practical help like looking after the DCs, run her a bath and get her to eat. Be prepared to listen to her rant and rave or sob and cry. Encourage her to see a counsellor. Make enquiries for her and try to lift the load of all the myriad of stuff we all need to keep the show on the road each day - even paying a bill or doing the shopping can seem a mountainous chore when one is in shock.
When her and her H need to talk, take the kids away for some long uninterrupted time. Give them lots of cuddles too, as their world will be different at the moment.
You may want to rip your BIL's head off, but resist the urge to slate him, either behind his back or to his face. Understand that it's going to be difficult for her to go from loving to hating him this quickly.
If your sis is an analytical person who needs to make sense of the past few months, encourage her to timeline when she last felt truly happy. Help her identify why she was feeling unsettled and uneasy - this really helps.