When my mother introduces my mother-in-law, she would say "This is ___". I know English hasn't many relationship terms, so is it just "This is my daughter-in-law's mum"?
English does not have such a word. Your mother would say: "This is Judy, my daughter's mother-in-law", or "This is Judy -- she's the mother of my son-in-law Bill."
There are terms in Yiddish and Hebrew, which might be used by older American Jews (such as my mother, who is in her 80s and the daughter of immigrants):
- machatonister (for the mother-in-law of your child)
- mechuten (for the father-in-law of your child)
- machatonim (plural)
The "ch" is pronounced in the back of the throat, as in German. Unlike other Yiddish words, these have not been imported into English, and few Americans of any religious background would know them.