[Jellby]

Where and how is the "Times New Roman" family defined? Is it embedded in the ePUB? Is the italic font embedded too?

(If you are going to embed fonts, you'd probably want to embed fonts that are free or allow full embedding in their license)

This is what I mean:

@font-face {
  font-family: "Crom";
  src: url("../fonts/Crom_v1.1.ttf") format("truetype");
}
@font-face {
  font-family: "Crom";
  font-weight: bold;
  src: url("../fonts/Crom_v1.1.ttf") format("truetype");
}

h1, h2 {
  font-family: "Crom";
}

Note that, in this case, I define the bold "Crom" exactly as the regular "Crom", that is to ensure the same font is used, whether the reader would like to have it bold or not. In your case, you have to define the italic "Times New Roman" differently.

Note too that, even if you do everything perfect, readers can still have bugs.