Bitbucket apparently changed something, and I have been googling for answers but there are so many topics of logins failing to Bitbucket that I don't even know which to start with. Most of them are spread out over the past few years. I don't see anything recent-- as in, something that would break logins between March and today.
hg push is the command I am using to push changes. It fails with an HTTP 403; the only output is "abort: HTTP Error 403: Forbidden".
I am using the same settings I have for years. My push process worked as recently as March.
I can log into the website using my email and password, which is different than the username I used to use to push.
https://bitbucket.org/blog/sunsetting-mercurial-support-in-bitbucket
After much consideration, we've decided to remove Mercurial support from Bitbucket Cloud and its API. Mercurial features and repositories will be officially deprecated on July 1, 2020.
You are getting a 403 because Bitbucket no longer supports Mercurial. The July 1st 2020 cutoff date fits between your March 2020 and now (July 29th 2020) time window.