Please note: This only works Apple <-> Apple or Android <-> Android
You can use the backup service to manually copy your data to a new device as long as it's from an Apple device to an Apple device or an Android device to an Android device.
To do this first log into your account on both devices.
Next on the device with your latest data go to the backup service and perform a backup.
Then open the app on your new device and go to the backup service and use the restore feature.
Your data should be copied to the new device.
It's very important that you only use backup on the device that has the latest information, and only use restore on the device that you wish to copy it to.
Please note: Restoring overwrites all data, it does not merge. This means you can not make changes on both devices independently.