What is the ruling on a wife secretly communicating with her husband's brother's wife without the knowledge of their husbands, for fear of causing problems with the husband's mother who disapproves of their communication, and is the husband's mother's stance of separating her children considered correct?

There is no objection to your relationship and communication with each other as long as it is within the bounds and regulations of the Sharia, especially if the relationship and love are for the sake of Allah and to strengthen the bonds between the spouses, provided that the costs of communication are from your own money, otherwise, with the permission of the spouses. It is advisable not to inform the mothers of the spouses about this to cut off the causes of animosity.