system_date_format_locale() has been removed in favor of loading formats with system_date_format_load($format_type). Language information is included the return value from system_date_format_load().
In most cases, you can just use format_date() directly.

Example:

Instead of:
$format = system_date_format_locale($langcode, $format_type);

Now do:

$date_format = system_date_format_load($format_type);
$format = isset($date_format['locales'][$langcode]) ? $date_format['locales'][$langcode] : $date_format['pattern'];
Introduced in branch: 
1.0.x
Introduced in version: 
1.0.0
Impacts: 
Module developers
Related Github Issues: 
https://github.com/backdrop/backdrop-issues/issues/709