Bitbucket is shifting away from app passwords towards the new Atlassian

API tokens. It's already impossible to create new passwords, and they

will stop working in June of this year.
API tokens work very similar to app passwords (both are fundamentally

just HTTP Basic auth), but there are two changes necessary:

Usernames are now e-mail addresses (where before they were explicitly

not e-mail addresses).
API tokens only work with the https://api.bitbucket.org/ endpoint, and

  not the https://bitbucket.org/api/ endpoint.

This change fixes up our form and API URL to handle this. Docs have been

updated, along with a big notice at the top.
I've also snuck in a fix for the GitHub PAT instructions to open the

link in a new tab.

Added a new Bitbucket account using API tokens and verified that API

  calls worked as expected.
Ran unit tests.
Built the user manual.