From Oscar's posts I think we now have enough clues to deduce what happened, which in my opinion is the Martha now lives with a close Burr family member/friend in Norwich. I expect Tanya got custody, but then an event (mental health, long days at work without Jim to help look after Martha etc) caused Tanya to resort to give Martha to someone like her parents. I reckon she didn't offer custody to Jim, as it would be painful for Tanya to see Martha getting close to Sarah, but that is why Sarah was bitter, because Jim is getting hate when he didn't have a say. Also, when Tanya feels ready, she can take back custody.
In theory, a dog being rehomed to your parents (temporarily or not) isn't a huuuuge deal. Which is why I reckon Oscar didn't think it was a big deal so originally thought he could joke about it, and knows she is well. I know several people who bit off more than they could chew (or situations changed) and gave their parents the dog instead (not great behaviour, but still relatively common). I think Jim is 100% pissed at the situation, but loves the "me and my ex wife are best pals!" narrative they have going on, but also potentially he understands her reasoning (maybe her mental health was really bad). I also think that we don't see pictures of Martha from while Tanya/Jim are in Norwich, because they know people like us would be able to tell those pictures were taken in Norwich only. Tanya doesn't want people to know she battled for custody and then gave her away, because that does look bad.