Name for the relationship of wife’s sister’s husband

I suspect they are, too - you'll be having co-son and co-son's mate next!

Seriously, you are talking about in-laws here.

From Wikipedia:

A brother-in-law (plural brothers-in-law) is the brother of one's spouse, the husband of one's sibling, the husband of one's spouse's sibling [relevant in the first case you mention], or the brother of one's sibling's spouse.

You can work out what sister-in-law covers.

Oh, and the perhaps unfamiliar terms used by Wikipedia:

sibling - a brother or sister

spouse - a husband or wife

... that is, they are hypernyms (like cutlery for knives, forks, ...)

No, there is no word for it in English.

I disagree with Edwin Ashworth and the Wikipedia article he quotes (which is entirely unreferenced). I would neither use nor normally understand brother-in-law to include a wife's sister's husband in modern English.

I would not be surprised to come across that usage in older writing (before the 20th century): many kinship terms were formerly used less precisely than today.

The OED says "Sometimes extended to the husband of one's wife's (or husband's) sister", but that entry has not been revised since 1888.

It is strange, and sometimes inconvenient, that there is no word.

I would call the man "my wife's brother-in-law." He's her brother-in-law, not yours.

"Brother/sister-in-law" is common U.S. English usage to refer to the spouses of your spouse's siblings. I have four brothers-in-law -- my sister's husband, my husband's brother, and my husband's sisters' husbands. And, I have three sisters-in-law - my husbands two sisters, and his brother's wife.