It is a common misconception that this diet causes weight loss. But first things first. She urgently needs to check her vitamin D status, and should consider re-commencing breast feeding. Low vitamin D is closely correlated with depression. If it is low, she needs the usual mega-dose to get levels up urgently. It is also important to get some regular exercise, preferably outdoors, as this helps enormously with depression, and will also assist with getting adequate vitamin D. It might also be worth looking at getting some counselling regarding the depression, and I am not sure to what extent she has help with raising the twins, but she certainly sounds like she could do with some.
Recent research confirms that breastfeeding is protective for the mother against further MS attacks while she is breastfeeding. It will also help her babies get their vitamin D levels up once she takes the megadose, through the breast milk. That will reduce the risk of the babies ever getting MS, and once the children stop breastfeeding, vitamin D supplements should start as per the instructions on my website. These need to be given to the other child as well.
To the diet. She should stop the milk-based products straight away. She should concentrate on nuts (almonds, cashews, etc), seeds and grains, preferably uncooked (through which she will get adequate fats to help with stopping tiredness), particularly soy products such as soy milk, tofu and tempeh, as well as legumes like chick peas. There are lots of possibilities like soy shakes, soy smoothies, etc. This will provide plenty of protein. She should also have plenty of carbohydrate rich foods (even if they contain refined sugars if need be in the short term ), until her weight stabilises. Thus she needs lots of fruit, vegetables (like potatoes) and pasta.
You didn't mention whether she is taking any medication, but if she is on one of the interferons, she should stop. That could be contributing to the depression, and research shows that women who stop breastfeeding to go back on interferons actually have an increased risk of relapse.