Struggling to find any documentation on how to override Craft's HTML for the forgotten password email.
The documentation can be found here: https://craftcms.com/docs/templating/forgot-password-form
Part of the full templating documentation that starts here: https://craftcms.com/docs/templating/global-variables
The actual HTML side of things is determined by the template you're using, which you can specify in the control panel -> settings -> emails. I believe these settings are considered 'self documenting' in that there are explanations on that page as to what they do.
One thing worth noting is that IIRC you can't extend a layout template with that template, which is a little unfortunate. So I have one general layout for all my general emails and duplicate that fully into the system email template.