Praise be to Allah.
What is required with regard to fasting is to refrain from doing anything that breaks the fast from the break of dawn until the sun sets, because Allah, may He be exalted, says (interpretation of the meaning):
“And eat and drink until the white thread of dawn becomes distinct to you from the black thread [of night]”
What matters is the break of dawn; if someone is not certain that dawn has broken, then he may eat until he is certain.
Similarly, if he knows that the mu’adhdhin gives the call to prayer ahead of time, or he is not sure whether he is giving the call to prayer on time or ahead of time, then he may eat until he is certain. However, it is more appropriate for him to stop (eating and drinking) as soon as he hears the adhaan, so as to be on the safe side.
Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him) was asked: What is the shar‘i ruling on the fast of someone who hears the adhaan of Fajr but continues to eat and drink?
He replied: What is required of the believer is to refrain from things that break the fast, such as eating, drinking and so on, when it becomes clear to him that dawn has broken, if the fast is obligatory, such as the fast of Ramadan, or fasting in fulfilment of a vow or as an act of expiation, because Allah, may He be glorified and exalted, says (interpretation of the meaning):
“And eat and drink until the white thread of dawn becomes distinct to you from the black thread [of night]. Then complete the fast until the sunset”
So if he hears the adhaan and knows that this is the call for Fajr, then he must stop (eating etc.)
If the mu’adhdhin gives the call to prayer before dawn breaks, then it is not obligatory to stop eating etc., and it is permissible to eat and drink until he is certain that dawn has broken.
If he does not know whether the mu’adhdhin gives the call to prayer before dawn has broken or after, then it is better and more prudent to stop eating etc. when he hears the adhaan, but it does not matter if he eats or drinks something whilst hearing the adhaan, because he does not know (for sure) whether dawn has broken.
It is well-known that people who live in cities where there is electric lighting cannot know whether dawn has broken by looking at the sky at the time of dawn; but he should err on the side of caution by paying attention to the adhaan and the timetables which state the time of dawn, mentioning hour and minute, acting in accordance with the words of the Prophet (blessings and peace of Allah be upon him): “Leave that which makes you doubt for that which does not make you doubt”; and “Whoever is cautious with regard to ambiguous matters will protect his religious commitment from shortcomings and will protect his honour from slander” and Allah is the source of strength.
End quote from Fataawa Ramadaan, compiled by Ashraf ‘Abd al-Maqsood (p. 201)
Based on that, if your mother and your brother were following someone whom they trust, who says that the adhaan is given ahead of the time of the true dawn, so they ate based on that, then there is no blame on them for what they did in the past, and their fast is valid.
But in the future, they should stop eating when the adhaan begins, because the issue of errors in timetables is very controversial, especially with regard to the degree of error. So the most prudent approach with regard to fasting is to stop eating when the adhaan begins, and the most prudent approach with regard to the prayer is to delay it for twenty minutes or more.
For more information, please see the answers to questions no. 26763 and 66202.
And Allah knows best.