Praise be to Allah.
What you must do is strive to resolve problems that may arise between your mother and your wife in a wise manner, so that you do not make your mother angry or wrong your wife. Yes, the rights that the mother has over her child are not like the rights that anyone else has over him, either his wife or his son, but that does not mean that others do not have rights over you, and the fact that you pay attention to the rights that your mother has over you does not mean that you should neglect the rights of others; rather you should give each one who has rights his or her rights.
The more you can do to provide separate accommodation for your wife and your mother, and reduce the opportunities for them to mix, the better, for that will be more effective in leading to a stable life and reducing problems. Please see the answer to question no. 167997
It is not permissible for a Muslim to accuse his fellow Muslim or cast aspersions upon his honour merely on the basis of suspicion. Allah, may He be exalted, says (interpretation of the meaning):
“O you who believe! Avoid much suspicions, indeed some suspicions are sins. And spy not, neither backbite one another. Would one of you like to eat the flesh of his dead brother? You would hate it (so hate backbiting) . And fear Allah. Verily, Allah is the One Who accepts repentance, Most Merciful”
Al-Bukhari (5144) and Muslim (2563) narrated from Abu Hurayrah (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: “Beware of suspicion, for suspicion is the falsest of speech. Do not spy on one another; do not seek out one another’s faults; do not hate one another; and be, O slaves of Allah, brothers.”
Al-Hafiz Ibn Hajar (may Allah have mercy on him) said in Fath al-Bari (10/486):
The prohibition applies to suspicion regarding a Muslim who is sound in his religious commitment and honour. End quote.
Al-Qurtubi (may Allah have mercy on him) said:
What distinguishes the suspicion that must be avoided from other kinds of suspicion is that every suspicion for which you know of no clear sign or obvious reason is haram and must be avoided. This applies if the person who is suspected is one of those who are known for dignity and righteousness, and is apparently trustworthy. Suspecting evil and treachery on his part is haram. This is in contrast to one who is well known among people for indulging in suspicious activities and who openly commits evil deeds.
End quote from Tafseer al-Qurtubi (16/331)
Ash-Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
Thinking of others may be divided into two categories:
The first category is thinking well of others, which requires you to think positively of your brothers so long as they are deserving of that. This applies to the Muslim who appears outwardly to be of good character; such a person is to be thought well of and praised for what we can see of his commitment to Islam and good deeds.
The second category is thinking badly of others (suspicion). This is haram in the case of a Muslim who appears outwardly to be of good character; it is not permissible to think badly of him or suspect him, as the scholars have clearly stated. They (may Allah have mercy on them) said: It is haram to think badly of a Muslim who appears outwardly to be of good character. With regard to thinking badly of someone for whom circumstantial evidence indicates that he deserves to be thought of in such a manner, there is no blame on the individual if he thinks badly of him. Hence a popular proverb says: “Be on your guard against people by thinking badly of them.” But this is not applicable in all cases, as is well known. Rather what is meant is be on your guard against people who deserve to be badly thought of, so do not trust them. A person will inevitably find thoughts coming to mind reflecting suspicion of anyone concerning whom circumstantial evidence justifies that, either because there are signs of that in his face, as you can see a frown and displeasure on his face at meeting you, and the like, or because of his situation and what people know about him or the things that he says, so he is ill thought of. In such a case, if there is circumstantial evidence to suggest it, there is no blame on a person if he thinks badly of him or suspects him.
End quote from Tafseer Soorat al-Hujurat (49)
To sum up:
Your wife is not sinning if she thought that about your mother, so long as your mother is known for that or there is circumstantial evidence to that effect. In that case, what you must do is keep them apart in a wise manner. If your wife has any proof for what she is saying, then try to find out the truth from your mother using wise means and tricks.
But if she does not have any proof and your mother denies that she took it, then there is nothing wrong with you asking her to swear an oath to that effect. In fact this is something that is prescribed in Islam. The shar‘i principle is that proof is required from the one who makes the claim and the oath is required from the one who denies it. In that case, there is no blame on you, in sha Allah, for what you did of asking your mother to swear an oath. In fact this may reduce the possibility of problems and resentment between them. But if you feel that your wife is telling the truth and that your mother’s feelings may be hurt by that, then try to compensate her for it by showing kindness and using tricks, and try harder with your mother to encourage her to repent to Allah, may He be glorified and exalted, and to stop doing such things. Try harder to suffice her and take care of her so that she has no need to do such things.
For information on the ruling on swearing on the Mushaf, please see the answer to question no. 98194
And Allah knows best.