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)
I am unsure whether you're asking who has the copyright or what you can do with the code.
Assuming this isn't a work-for-hire and assuming the copyright hasn't been transferred, the code belongs to the person or people who wrote it, so presumably to the author.
Since you apparently have no license, you can't do anything with the code, except read it in place.