What is the ruling on praying 'Isha in mosques, given that the Umm Al-Qura calendar delays its time by approximately 10 minutes compared to the mosques' establishment of the prayer?

If the calendar is approved by a reliable authority, then act upon it, and its difference from another calendar will not harm you. Most calendars in Islamic countries are issued under the supervision of Ministries of Endowments and are reliable. It is advisable not to doubt the prayer times issued by reliable authorities. If the calendar is not approved by a reliable authority, then act according to the later calendar in time, because the basic principle is that the time has not yet entered, and one should not pray with doubt regarding its entry.