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 Gmail.com, iCloud.com, Exchange, or any other cloud service.
To force your contacts to appear at iCloud.com immediately, simply go to your iPhone Settings > Contacts > Accounts > 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.