Please see the scenarios steps.
1. Project is created with task assignments and is published to the Project Server
2. Task updates are working fine from project server.
3. If you update the Health of a project from project professional and publish the project, updated health information is shown under My task.
4. If you try to update the Health from My task and submit the updates, the task approver gets the updates with all the changes including the Health information. Once he approves and publish the project, the Health information is not updated in the Project Professional.
Thanks for helping me in this issue.
Junaid