Is it permissible to individually collect money allocated for 'Umrah (Sunnah) for going on Hajj (Fard), knowing that this will cause the wife, who has no income, not to go, and will this action lead to psychological repercussions for her?

Hajj becomes immediately obligatory if a Muslim is able. Ability includes provisions and a mount beyond what is needed for the expenses of those he supports. This ability is considered at the time the people of the land set out for Hajj. Based on this, you are not obligated to save for Hajj as long as you are not able. It is permissible for you and your wife to use the accumulated money for 'Umrah. Your wife's Hajj expenses are not obligatory upon you unless it is a voluntary contribution. As for the amount collected for your wife for 'Umrah, if it was not a seized gift, you may reclaim it.