There should be a way to export and import (migrate) issues from Bitbucket Cloud Issues into Jira Software Cloud.
You can export issues from Bitbucket as JSON, and you can import issues into Jira as JSON, but the two formats are not compatible so the import will fail.
Jira Cloud has an equivalent Bug for this issue:
- In Bitbucket Cloud go to Settings > Issues > Import & export.
- Press Start export button.
- Download JSON when prompted.
- In Jira Software Cloud go to Jira settings > System > External System Import.
- Select JSON importer.
- Upload JSON when prompted.
Issues should successfully import into Jira.
Import fails as the format is incompatible.
JSWCLOUD-16239 on importing to Jira Server and migrating from there.
Another option is to write your own script to transform the JSON from Bitbucket into the format accepted by Jira.