This Code can be placed in the site settings or page settings custom code area:

<script>
  console.clear();
  /* globals
  $
*/

  const data = {
    months: {
      en: [
        'January', 'February', 'March', 'April', 'May', 'June',
        'July', 'August', 'September', 'October', 'November', 'December',
      ],
      local: [],
    },
    days: {
      en: [
        'Monday', 'Tuesday', 'Wednesday', 'Thursday', 'Friday', 'Saturday', 'Sunday',
      ],
      local: [],
    }
  };

  data.months.local = [
    'Januar', 'Februar', 'März', 'April', 'Mai', 'Juni',
    'Juli', 'August', 'September', 'Oktober', 'November', 'Dezember',
  ];
  data.days.local = [
    'Montag', 'Dienstag', 'Mittwoch', 'Donnerstag', 'Freitag', 'Samstag', 'Sonntag',
  ];

  if (data.months.local.length !== 12) {
    console.error('Months are incorrect! Check your script.')
  }
  if (data.days.local.length !== 7) {
    console.error('Days are incorrect! Check your script.')
  }

  const shortenDaysMonths = daymonth => daymonth.substring(0, 3);
  const convertToLocal = (daydate, whatToConvert) => {
    whatToConvert.each(function() {
      const theObject = $(this);
      let text = theObject.text();

      if (daydate === 'm' || daydate === 'month' || daydate === 'months') {
        for (let i = 0; i < data.months.en.length; i++) {
          text = text.replace(data.months.en[i], data.months.local[i])
          text = text.replace(shortenDaysMonths(data.months.en[i]), shortenDaysMonths(data.months.local[i]))
          theObject.text(text)
        }
      } else if (daydate === 'd' || daydate === 'day' || daydate === 'days') {
        for (let i = 0; i < data.days.en.length; i++) {
          text = text.replace(data.days.en[i], data.days.local[i])
          text = text.replace(shortenDaysMonths(data.days.en[i]), shortenDaysMonths(data.days.local[i]))
          theObject.text(text)
        }
      }
    });
  };
</script>

On each page where you need to translate dates, you'll have to add this script in the page settings:

<script>
  const allDates = $('.date-month');
  const allDays = $('.date-day');

  convertToLocal('m', allDates);
  convertToLocal('d', allDays);
</script>