Praise be to Allah.
If a person misses the fast of 'Ashura, it cannot be made up because there is no proof for that, and because the reward is connected to fasting on the tenth day of Muharram, and that time has passed.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: If the day of 'Ashura comes and the woman is menstruating, can she make up the fast? Is there a principle concerning naafil acts of worship that may be made up and those that may not be made up? May Allah reward you with good.
He replied: Naafil acts of worship are of two types: those for which there is a reason and those for which there is no reason. Those for which there is a reason end when the reason ends, and they cannot be made up. An example of that is tahiyyat al-masjid (two rak‘ahs to “greet the mosque”). If a man comes and sits down, and sits for a long time, then he gets up wanting to pray tahiyyat al-masjid, it will not be tahiyyat al-masjid, because that is a prayer that is done for a reason and is connected to the reason; once the reason no longer applies, it is no longer prescribed. Another example, it would seem, is the day of ‘Arafah and the day of 'Ashura. If a person delays fasting the day of ‘Arafah and the day of 'Ashura, with no excuse, there is no doubt that it cannot be made up and it would not benefit him if he did make it up, i.e., it would not benefit him in the sense of it being the day of ‘Arafah or the day of 'Ashura.
But if that day comes when a person has an excuse, such as the woman who is menstruating or bleeding following childbirth, or a person who is sick, then it seems that it cannot be made up in this case either, because it is connected to a specific day and the ruling ceases to apply once that day is past. From Majmoo‘ Fataawa Ibn ‘Uthaymeen, 20/43
But if a person is excused for not fasting – such as women who are menstruating or bleeding following childbirth, or those who are sick or are travelling – and he usually fasts on that day or he had the intention of fasting that day, then he will be rewarded for his intention, because of the report narrated by al-Bukhaari (2996) from Abu Moosa (may Allah be pleased with him) who said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “If a person falls sick or travels, he will have a reward recorded for him like that of what he used to do when he was at home and was healthy.”
Ibn Hajar (may Allah have mercy on him) said: “He will have a reward recorded for him like that of what he used to do when he was at home and was healthy” means that he is like one who used to do an act of worship then was prevented from doing it, but he had the intention that were it not for the obstacle he would have continued to do it. Fath al-Baari.
And Allah knows best.