Is it permissible to pressure a mother to move and live with her son in the city to care for her sick father, especially if the son is their sole provider?

It is not an act of righteousness towards your mother to ask your maternal uncle to pressure her. Rather, it is obligatory to treat her with gentleness, kindness, respect, and humility. There is no harm in appealing to her through someone who can intercede with her, and in striving to convince her and explain your need for her help. If these means are effective, then good; otherwise, let her reside in the place that pleases her. As for your father, if it is possible for him to live with you in the city, such that you have someone to attend to his affairs, and neither he nor your mother are harmed by their separation, then there is no objection to bringing him with you, while being careful to provide for your mother's needs. If this is not possible, then leave him with your mother and provide them with whatever care and attention is within your means.