What is the way to deal with the constant quarreling between the wife and the mother, which has led the wife to consider divorce, given the questioner's desire not to lose either of them?

Your duty is to honor your mother and provide an independent dwelling for your wife, as this is her right, and she is not obligated to accept living with your mother. Be wise in combining these two matters, and strive to mend the relationship between them. Explain to your wife that her kindness to your family brings immense reward. If she refuses shared accommodation, then provide her with an independent dwelling, even if it's just a room with its amenities. If that is difficult, live close to your mother. Do not wish for death because of disagreements; rather, ask Allah for well-being and seek His help.