After the projects are added to multiple programs, you may need to go back to Project detaisl page and select the programs manually.
Else, you may automate through a DB trigger/ report to sync the programs in the project details whenever the project is added to any program. And execute a one time update to populate the same for existing mappings.
If you are on lower version, then probably a DB script will help you. If you add a 'Program Associations' portlet on your project overview page, it will show you the updated programs... but the 'Associated Programs' field on the project details doesnt really update and that is why you will need to run DB scripts.
I had faced similar issue and raised a case with HPE..