What is the duty of someone who caused two people to break their fast by imitating the sound of the Adhan? And is it obligatory for him to fast sixty days for each of them, and is it permissible to postpone the fasting of one of the sixty days until after the next Ramadan?

You did an inappropriate thing by imitating the sound of the Azan at the time people were breaking their fast. If you did not intend mischief or deception, then there is nothing upon you, but learn to appreciate situations. If you intended to deceive the fasting people, then you must repent and seek forgiveness, and no expiation is required of you.

As for your mother and uncle, there is no sin upon them for breaking their fast, thinking that Maghrib had entered. Scholars differed on whether they are obliged to make up that day: the majority believe in making it up, while some scholars (such as Sheikh Al-Islam Ibn Taymiyyah and Sheikh Ibn Uthaymeen) believe that making it up is not obligatory because they are excused. However, the more cautious approach and to avoid disagreement is to make up the day, and this is the opinion of Sheikh Ibn Baz.