'not so long after' can mean within a few minutes, hours, days months or even years depending on the situation. For example: If you were talking about the history of a person(their life story) it could mean a few years later. If you were talking about a situation all happening within one day, it could mean hours. If you were talking about part of a day, it could mean minutes. If it happened seconds or minutes after we would usually use 'immediately after'.

"just then" is a very abstract expression of time. It means nothing out of context. "not so long after" is the correct phrase.

Not so long afterwards, [ soon] my mother came home.