OK, I've managed to figure it out only partially.
I added the following at the end of the themes/delta/footer.php file.
<script type="text/javascript">
$.datepicker.regional['pl'] = { // Polish regional settings
closeText: '<?php echo osc_esc_js ( __('Zamknij') ); ?>', // Display text for close link
prevText: '<?php echo osc_esc_js ( __('Poprzedni') ); ?>', // Display text for previous month link
nextText: '<?php echo osc_esc_js ( __('Następny') ); ?>', // Display text for next month link
currentText: '<?php echo osc_esc_js ( __('Dziś') ); ?>', // Display text for current month link
monthNames: ['<?php echo osc_esc_js ( __('Styczeń') ); ?>','<?php echo osc_esc_js ( __('Luty') ); ?>','<?php echo osc_esc_js ( __('Marzec') ); ?>','<?php echo osc_esc_js ( __('Kwiecień') ); ?>','<?php echo osc_esc_js ( __('Maj') ); ?>','<?php echo osc_esc_js ( __('Czerwiec') ); ?>','<?php echo osc_esc_js ( __('Lipiec') ); ?>','<?php echo osc_esc_js ( __('Sierpień') ); ?>','<?php echo osc_esc_js ( __('Wrzesień') ); ?>','<?php echo osc_esc_js ( __('Październik') ); ?>','<?php echo osc_esc_js ( __('Listopad') ); ?>','<?php echo osc_esc_js ( __('Grudzień') ); ?>'], // Names of months for drop-down and formatting
monthNamesShort: ['<?php _e('Sty'); ?>', '<?php _e('Lut'); ?>', '<?php _e('Mar'); ?>', '<?php _e('Kwi'); ?>', '<?php _e('Maj'); ?>', '<?php _e('Cze'); ?>', '<?php _e('Lip'); ?>', '<?php _e('Sie'); ?>', '<?php _e('Wrz'); ?>', '<?php _e('Paź'); ?>', '<?php _e('Lis'); ?>', '<?php _e('Gru'); ?>'], // For formatting
dayNames: ['<?php echo osc_esc_js ( __('Niedziela') ); ?>', '<?php echo osc_esc_js ( __('Poniedziałek') ); ?>', '<?php echo osc_esc_js ( __('Wtorek') ); ?>', '<?php echo osc_esc_js ( __('Środa') ); ?>', '<?php echo osc_esc_js ( __('Czwartek') ); ?>', '<?php echo osc_esc_js ( __('Piątek') ); ?>', '<?php echo osc_esc_js ( __('Sobota') ); ?>'], // For formatting
dayNamesShort: ['<?php _e('Nie'); ?>', '<?php _e('Pon'); ?>', '<?php _e('Wto'); ?>', '<?php _e('Śro'); ?>', '<?php _e('Czw'); ?>', '<?php _e('Pią'); ?>', '<?php _e('Sob'); ?>'], // For formatting
dayNamesMin: ['<?php _e('Ni'); ?>','<?php _e('Po'); ?>','<?php _e('Wt'); ?>','<?php _e('Śr'); ?>','<?php _e('Cz'); ?>','<?php _e('Pi'); ?>','<?php _e('So'); ?>'], // Column headings for days starting at Sunday
weekHeader: '<?php _e('Tydz'); ?>', // Column header for week of the year
dateFormat: 'dd.mm.yy', // See format options on parseDate
firstDay: 1, // The first day of the week, Sun = 0, Mon = 1, ...
isRTL: false, // True if right-to-left language, false if left-to-right
showMonthAfterYear: false, // True if the year select precedes month, false for month then year
yearSuffix: '' // Additional text to append to the year in the month headers
};
$.datepicker.setDefaults($.datepicker.regional['pl']);
</script>
And I see the Polish names now. But when I switch to English, I still see the Polish names. So it looks like it doesn't follow locales.