It may take some time for changes to automatically sync to your contacts (up to 8 hours).
This said - here is how to force the app to sync on iPhone and Android:
Go to the app's Settings screen
Toggle Contact Syncing from On to Off
Toggle Contact Syncing back to On
All changes should update in your phone contacts.
Note: If you set up contact syncing on your iPhone using the CardDav legacy method, please try the steps below:
Go to your iPhone Settings
Tap Mail, Contacts, Calendars
Tap Haystack
Toggle Account from On to Off
Toggle Account back to On