Your friend emigrated a couple of years ago to the other side of the world with her very difficult husband. He makes it clear that the house and child care is her responsibility and does nothing to support her (except finacially as he earns more than her). Her first DD is 4 and her second DD is 1 month.
DD1 is EXTREMELY difficult. She was a delight before she emigrated, very bright and bubbly and interested in everything but really well behaved too. When you went to see them about a year after they emigrated you were shocked by the change in DD1. She was demanding, refused to do anything for her Mum, defiant and basically was allowed to get away with murder.
You say your friend and her difficult husband try to control DD1 with empty threats. You're concerned that the child's diet isn't as good as it was at home. She is behind at school. Her behaviour seems oppositional and she has a very short attention span, though you say she's affectionate with her new baby brother.
Your friend has Skyped you, in a state of high anxiety and tears, saying she can't cope.
Personally I think it's reasonable for a 4yo to be showing signs of distress after a major house move and a new addition to the family. She would need far more loving affirmation, not threats & criticisms, and rewards for observing safe boundaries rather than getting (negative) attention by misbehaving so as to be yelled at.
The child may turn out to have ADD/ODD but it's unfair to suppose a learning disability when she's been going through such massive disruptions to her life. Any child would react badly. Combined with the fact she should be pushing her limits at this age, it's to be expected really. From what you've posted, the problem seems to be in the parents' responses rather than the little girl herself.
Does your friend feel isolated and unsupported since the move? What resources has she got, and what else could be found? How much more "difficult" has the husband become since they moved?