Praise be to Allah.
One of the conditions of prayer being valid is the body and clothes being free of impurity. If any impurity (najaasah) gets on to the body or clothes, it is essential to remove it before starting to pray.
Your mother can set aside clothes for prayer and clothes for work, then when she wants to pray she could take off the work clothes and put on the other clothes.
Or it is enough for her to wash the place where the impurity is on the clothes, then pray wearing them if she wants.
Wiping the head with water during wudoo’ is one of the obligatory parts of wudoo’. But if a woman is wearing a head cover, she does not have to take it off and she may wipe over it, especially if it is difficult to take it off. Al-Bahooti (may Allah have mercy on him) said: Wiping over a woman’s head cover is valid, especially one that is wrapped around the chin, because Umm Salamah used to wipe over her head cover, as was mentioned by Ibn al-Mundhir, and the Prophet (blessings and peace of Allah be upon him) said: “Wipe over the khuffayn (leather slippers) and the head cover.” Narrated by Ahmad. And because it is a covering that is difficult to remove.
End quote from Daqaa’iq Ooli al-Nuha, 1/62. See also al-Mughni, 1/187
The hadeeth “Wipe over the khuffaayn and the head cover” was classed as da‘eef (weak) by Shaykh al-Albaani (may Allah have mercy on him). See Da‘eef al-Jaami‘, no. 1270.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: Is it permissible for a woman to wipe over her head cover?
He replied: The well-known view of the madhhab of Imam Ahmad is that she may wipe over the head cover if it is wrapped around the chin, because that was narrated from some of the women of the Sahaabah (may Allah be pleased with them).
Whatever the case, if there is any difficulty involved, either because it is cold or because it is difficult to take it off and put it on again, that there is nothing wrong with wiping over it in this case, although it is better not to do so.
End quote from Majmoo‘ Fataawa Ibn ‘Uthaymeen, 11/171.
The basic principle is that it is not permissible to delay prayer until the time for it is over, except when one has an excuse, because Allah says (interpretation of the meaning): “Verily, As‑Salaah (the prayer) is enjoined on the believers at fixed hours” [al-Nisa’ 4:103]. This has been discussed in the answer to question no. 21958.
What is required of the sister who is asking this question is to take all possible measures to offer the prayers on time. If that is too difficult for her or she is not able to do it because of the way her work is organised, then there is nothing wrong with her putting two prayers together. So she may put Zuhr and ‘Asr together, and put Maghrib and ‘Isha’ together, at the time of the earlier or later prayer, according to what is easiest for her.
The scholars of the Standing Committee for Issuing Fatwas were asked: Is it permissible to delay prayer until the time for it has ended, such as ‘Asr prayer, in the case of necessity, namely if a doctor is doing surgery and if he leaves the patient, even for a short time, that will pose a danger to his life?
They replied: The doctor who specialises in doing surgery should take care to do surgery at times when he will not miss out on offering the prayers at the proper time. In cases of necessity it is permissible to put two prayers together at the time of the earlier or later prayer, such as putting Zuhr and ‘Asr together, or Maghrib and ‘Isha’, according to what is dictated by necessity. End quote.
Fataawa al-Lajnah al-Daa’imah, 25/44
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked about the ruling on delaying prayer until the time for it is over because of work, such as a doctor who is on call.
He replied: Delaying prayer until the time for it is over because of work is haraam and is not permissible, because Allah says (interpretation of the meaning): “Verily, As‑Salaah (the prayer) is enjoined on the believers at fixed hours” [al-Nisa’ 4:103]. And the Prophet (blessings and peace of Allah be upon him) defined the set times for prayer. So whoever delays the prayer until after these times or does it before these times has transgressed the limits set by Allah, and those who transgress the limits set by Allah are the wrongdoers. So it is not permissible for a person to delay the prayer until the time for it is over for any kind of work. But if the prayer is one that may be put together with the one that comes before it or after it, and it is difficult for him to offer each prayer at its own time, then he may put the prayers together, such as if his shift at work is during the time for Zuhr, and it is too difficult for him to pray Zuhr on time, then he may put it together with ‘Asr prayer, and by the same token he could put ‘Isha’ prayer together with Maghrib, because it is proven in Saheeh Muslim that Ibn ‘Abbaas (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) put Zuhr and ‘Asr, and Maghrib and ‘Isha’, together in Madeenah when there was no state of fear or rain. They asked Ibn ‘Abbaas: What was the purpose behind that? He said: He did not want to make things difficult for his ummah -- i.e., he wanted to avoid making things difficult for them by not putting prayers together. End quote from Majmoo‘ Fataawa Ibn ‘Uthaymeen (12/33).
And Allah knows best.