You might get some use out of the stash plugin as BitBucket is based on Stash code.
What Atlassian had to say:
We have unified our Git products under the Bitbucket name. With
Bitbucket, now you have a range of options that can be adopted by teams
of all sizes and requirements: Bitbucket Cloud (previously known as
Bitbucket), Bitbucket Server (previously known as Stash) and Bitbucket
Data Center (previously known as Stash Data Center).
About the plugin:
This plugin notifies an Atlassian Stash instance of Jenkins builds in
progress and of their results. The build results will show up in the
commit list in Stash or in a Stash pull request and provide immediate
and easy to scan feedback about a commit. It is possible to navigate
from build result information in Stash directly to the build in
question. The plugin can be configured on a per-job basis, globally
for all Jobs or both to minimize the configuration overhead.