Who is the owner of code uploaded from a personal account to a private Bitbucket repository and there is no license defined in the code, or there is only the name of the author in the header of the code? (based on that there is no any signed contract between the repository owner and the code owner)
The actual creator(s) of the code own the copyrights, unless a written contract transferring the copyright was signed, or unless the work was done by an employee (or employees) within the scope of his or her or their employment, or Unless the code was a "work-made-for-hire" and is so specified in a written contract signed by the creator(s) that confirms with the jurisdiction's law.
This is always true, no matter who uploads code, or what repositories it may come from or be sent to. This is basic copyright law and is true in all Bern Convention countries, except that the provisions on employer ownership may vary somewhat. They are correct for US law.
It is best practice to always include a copyright notice or statement in code being released or placed in a repository, to make it clear just who owns the rights, even when this is not required. Some open source licenses require such a statement.
It does not really make sense to upload code to a repository without a license, except perhaps as an educational example that may not be reused. Only the rights-holder, or someone with permission from the rights-holder,(including an appropriate license) should upload such code in the first place.