Using the Twilio API I can get message details from its SID.
curl -X GET 'https://api.twilio.com/2010-04-01/Accounts/{YourAccountSID}/Messages/{MessageSID}.json' \
-u '{YourAccountSID}:{YourAuthToken}'
However, this does not return the SendAt date submitted for messages that are in the scheduled state, as shown in the Twilio UI.
What API call do I need to make to retrieve the time the future message will be sent?
From their scheduling API docs, it doesn't look like that field is exposed. Since SendAt is required to create a scheduled message, you could just store that info elsewhere at time of message creation instead of querying Twilio's API to get it.