While Contact Mover does sync contacts between accounts on your iPhone in just seconds, it does depend on iOS to then transfer those changes to the cloud, and iOS sometimes stalls. As a result, sometimes there is a delay before seeing your contacts at contacts.google.com, iCloud.com, Exchange, or any other cloud service.
If contacts aren't appearing at iCloud.com:
To force your contacts to appear at iCloud.com immediately, simply go to your iPhone Settings > Contacts > Accounts > iCloud > iCloud, and turn off Contacts temporarily. Choose to keep your contacts when prompted. Wait 30 seconds, and then turn Contacts back on. Now your contacts should appear at iCloud.com shortly.
If contacts aren't appearing at contacts.google.com:
For the specific task of syncing Google and iPhone contacts, I recommend using my specialized app for syncing Google contacts called "Contacts Sync" as it does not depend on iOS to transfer changes to Gmail.com. It can also sync groups. Here is a link to the app:
For any other issues, please email support.