I followed the instructions on this page
https://support.atlassian.com/bitbucket-cloud/docs/add-an-api-token-to-sourcetree-or-another-application/
I am running SourceTree version 3.4.31 on Windows.
One discrepancy is that at the point of pasting the Token. Rather than a field, I see a button "Refresh API Token" When I click that I have to paste the token into the password field.
After I do that, I see a "Authentication OK" status.
However when I then try to push some changes, it fails with a message that use of passwords is deprecated.
If I wait for a while, things work, but then days later it fails again.