How my Keep Safe data can be transferred to another Apple phone with Keep Safe (used by my mother) automatically?
The software features an option called Private Cloud which allows you to upload the data to access it on all the devices where the software is installed. Use the Private Cloud to migrate the data, then log in on the second device and access the Private Cloud and that's it. You can now begin transferring the data from one device to another through the Private Cloud.
Browse the app page to see more details about the features and supported options: https://itunes.apple.com/us/app/keep-safe-private-photo-vault/id510873505?mt=8