If you don't have these fancy fonts, then it is highly likely that your correspondents won't have them either. Is it worth it?

Other email clients have different fonts. Not all my contacts use Thunderbird and I see their signatures just fine even though I don't have their fonts.

So, name a few of these fonts then tell us if they are or are not installed on your own computer(s).

It may be that for reasons I don't pretend to understand, you do have some fonts which Thunderbird chooses not to include in its formatting tools. Collecting the names of such fonts may help us detect what limitations are being applied.

Alternatively your correspondents may be using something like Google Fonts where you provide an URL and the font is downloaded on demand. If you're happy about generating your own html markup you should be able to include these in your own messages. As in these matters generally, it needs consent on both parts. If you use these online fonts then you may find that cautious correspondents may not like their computers downloading any content from random locations on the Web and block such attempts.

Either way, my own view is that email is not wysiwyg, you have no certainty that your correspondent will see it as you wish, and so it's not really worth the effort. I have a colleague who does something similar to what you're proposing, but I only see his cursive font on certain platforms and clients.

I downloaded "The Woodlands" font and it installed in Open Office and Microsoft Office. How do I install it in Thunderbird?

You didn't have this font. So, nor will most of your correspondents. So why would you expect this font to appear in your correspondents' email clients? It wasn't on your computer until you downloaded it, so they would have to do the same.

I don't think it is going to work. I'd say that Thunderbird is trying to protect you from your misguided efforts.

I have just downloaded this font and installed it here in Linux. It appears in Thunderbird.

I suggest you research how to install it into Windows rather than into your office productivity applications.