Is it permissible for a wife to refuse to show her mother messages on her personal mobile phone, fearing that her mother might see her private matters, and would she be sinful for refusing, given her feeling of distress about it?

A mother does not have the right to look at her daughter's phone messages without her permission. There is no blame on the daughter for preventing her mother from doing so, especially regarding messages from her husband. The daughter must be gentle and wise in dealing with her mother to avoid upsetting her, because the mother's right is great, and honoring her is among the most obligatory duties.