The apostrophe is not a letter. It is not pronounced. It is always silent. It cannot change a word’s pronunciation by its presence or absence.
Therefore the twins I know and the twins’ mother will pronounce the “twins” part of it in there exactly the same way.
The idea of twins’ having some sort of a /ˈtwɪnzəz/ pronunciation is not just that it is informal; it is not. It is wrong.
The word that would be pronounced that way would have to be something like twinses, which does not to my knowledge exist. But if it did, then adding an apostrophe to make it possessive via twinses’ would still leave it pronounced the same way. Apostrophes do not change a word’s pronunciation.