I've never heard this one but it certainly makes for an interesting piece of Pop Latino history. I'm not sure how popular the song was at the time but it certainly was successful enough to be recorded by a number of very well known performers (more on that later).
This novelty song was written by the iconic songwriter Jule Styne and the lyricist Herbert Magidsen (who won the first Oscar for Best Original Song in 1934 for the "The Continental").
The versions I've heard online seem to all trade equally in both Mexican and Irish linguistic and musical stereotypes (and New Jersey stereotypes in the Bing Crosby version above). But aside from Conchita's long name and the "ethnic" names of their children, the song is very matter-of-fact. This was certainly born out in the Jack Benny live broadcast where I first discovered the song. After the band and crooner finished, the crowd erupted in cheers. And that was it. There were no comments or patter about the song's themes (and the Jack Benny show was nothing if filled with groan-worthy patter).
The Glenn Miller version features vocals by Tex Beineke and the Modernaires in a pretty swingin' and broguing version of the song. I was also able to find a version by Dinah Shore that's worth a listen.