The closest actual synonym is perhaps
What is the sense of a henpecker?
It's a relationship where the woman "tells the man what to do" - she usurps the (supposed) dominating, leading role of the male by constantly telling the male what to do, by "getting the upper hand", by commanding situations.
A harridan is "any" "bossy old woman". If you were using this archaic word, you could use it to describe any (say) bossy female politician, bossy old widow, etc.
In contrast, a henpecker (or "ball-buster") is specifically a wife who eliminates the masculinity, authority of the husband, through constant nattering.
And a shrew, if you were to use that archaic word - for me a shrew is a mean / bad-tempered / etc "independent woman" who doesn't even want to get married or have anything to do with men.
Regarding words like "nag" or "cow" ...
A vulgar synonym of "henpecked" is "pussy-whipped"; the point is the male's (supposed, whatever) usual place of authority, of decision over day to day elements in the household, has been usurped by the female. So, while a henpecker wife may indeed be a nag, or a cow, she may indeed just be "loud" ... but then, conversely, she may be the "silent staring" type of henpecker! If you're really trying to describe the notion that she has "got it over" the husband - the husband is henpecked or "pussy-whipped" - about the only real synonym of "henpecker" I can think of is "ball-buster".
Or indeed perhaps simply "dominating wife" as a phrase.