I am looking for an expression or term that would describe the relationship between the two mothers of their married children.
There's a specific term for the relationship between people whose children marry each other: co-parents-in-law.
The relationship between people whose children marry each other: the parents of the bride vis-à-vis the parents of the groom.
The Wiktionary entry additionally includes usage notes stating that in normal conversation the generic "in-laws" may be used, although this could be ambiguous.
To avoid ambiguity in everyday speech you can rely on context, or use phrases such as "my son-in-law's mother" or "my daughter-in-law's mother".
Wiktionary also lists coordinate terms such as:
Co-mothers-in-law specifically answers your question:
The relationship between women whose children marry each other; the mother of the bride vis-à-vis the mother of the groom.
In the Spanish language this word exists. It is Consuegros. Sadly, there is no equivalent in the English language to describe the relation of the in-laws to each other. Perhaps a version of this, "Co-mothers-in-law." Example, "Now that our children have married each other, I suppose that makes Marie my co-mother-in-law."