Praise be to Allah.
It is obligatory to wash all the parts of the body that are specified for washing in wudu, in obedience to the words of Allaah (interpretation of the meaning): “O you who believe! When you intend to offer al-salaat (the prayer), wash your faces and your hands (forearms) up to the elbows…” [al-Maa’idah 5:6], and following the example and teaching of the Prophet SAWS (peace and blessings of Allaah be upon him) recorded in the saheeh ahaadeeth, which describe the parts of the body to be washed in wudu. If, after praying, a person finds something on these parts of his body that prevented water from reaching them, then he must remove it and repeat his wudu and prayer.
As far as cutting the nails is concerned, this does not mean that one has to repeat wudu, and in fact doing so is not encouraged.