This is very sad and a terrible shock for your poor sister & distressing for the children, but I agree with the comments that she needs to get on with things now. Six weeks is a long time, even longer in the poor children's eyes.
He has behaved awfully, men often do when they have young children, it's unforgivable.
Why have the dogs & hamster been taken from the home? Pets provide normality to a home, it's very difficult to be depressed when you have a demanding animal around. This would surely be upsetting for the children too, losing their pets. You all need to be pragmatic & practical. Tell her she needs to get up & plan for the children's future, assuming now that he's not coming back.
Doesn't she work? Many mothers work & this provides a focus & reason to carry on.
You say she's crying about the house & schools but she should be worrying about the impact this is having on the children & finding some resilience for them.
You say she won't be able to live off savings forever, no, she'll have to get a job & change things to support the children like millions of abandoned mothers do. Downsize, get a lodger, whatever.
The husband will undoubtedly reappear at some point. I would be over there looking for him to find out the truth behind his behaviour & to confront him about the future so she knows where she is & what to do next.
She should be cooking, doing all the normal things etc which will help her. Wallowing in bed for hours every day is ridiculous when you have three young children, normality is essential, she's making it worse for the poor children.
Don't get me wrong, it's a horrible shock, but reading between the lines he may have left her because:
a)She doesn't work & contribute to the household & he's tired of being the main breadwinner
b) she has zero resilience & is too dependent on him & he's met a strong, independent woman (possibly a single mother) & has made unfavourable comparisons, possibly fallen for her
There's no mention in your post about the relationship being good to begin with or any happy family times.
I feel there's more to this story. You say she can do one activity a day. This is because you're letting her do one activity a day. This enormous safety net isn't helping, not are the medications.
So she can access an Apple account, have a consultation with a lawyer, but not cook for her children or walk the dogs? People are surviving wars & atrocities in the world far worse than this right at this moment & surviving. Was she quite a weak person before all this? Did she have a good friendship network, activities/playdates for the kids etc? You don't say how old they are except that they are young, but they need both a semblance of normality & distractions right now.
The human spirit is strong when it needs to be & she needs to toughen up for her children, they are the priority here, seeing their mother completely fall apart is the worst thing she can do.