Hi @AbhijeetKumar,
Welcome to the Atlassian Developer Community.
Bitbucket Cloud (available at bitbucket.org) provides a POST /2.0/repositories/{workspace}/{repo_slug}/src REST API endpoint which can be used to create new commits in the repository by uploading files.
You can find the relevant documentation and some examples here.
I see that you mention that the repository can’t be cloned, however I’m wondering why that’s the case and if it wouldn’t be possible to change that so that the standard git commands can be used for updating the repository. Feel free to share a bit more about the use case as well.
Cheers,
Caterina