The column picklist is not being displayed when typing a table name followed by a period. The table is not owned by the current schema.
Verify under View | Toad Options | Editor | Code Assist -- 'Display picklist after typing object name followed by a period' is checked.
If a Public Synonym has been created for the table then ensure:
View | Toad Options | Editor | Code Assist |Toad Insight Objects | 'Public Synonyms' is checked
If tables are not synonyms, however, they are in a different schema, then prefix the table name with the schema name.
If a Private Synonym has been created for the table, change the Private Synonym to be a Public Synonym.
This issue is fixed in version 12.6 of Toad for Oracle. Please download the latest release here.