Every email client renders HTML in a slightly different way. Each one has their own in-built styles which are applied to an email when the user views it, and these often override the styles you have applied to your signature.
The best way to get control of the styles in your email, is to apply them directly in-line to each element in the signature.
So putting a "div" tag around your signature and adding the font styles directly to that will be more consistently successful than trying to use a "style" tag and defining styles within it. Links must be styled directly in the "a" tag for your style to be consistently applied.
Phone styles can be an exception see: Phone Styles