As discussed, implementation of support for Bitbucket private factories flow is split on the 2 steps:
First step implemented under CRW-1372 and provides minimal ability to use factories for private BB repos, with manual configuring of the credentials in the user's namespace by user or admin.

Second step is what this issue about, supposes avoiding of manual credential configuration, and in case of unauthenticated user, initiate OAuth flow, login redirects, automated request for personal access token creation via Bitbucket REST API, and storing this token in user-s namespace for further usage.