Performing salah and covering the head, they are both orders of Allah. When you are performing a salah, you are following an order that came from Allah; so just like it, you shall use a veil and obey him as well.
Fulfilling a fard and not fulfilling the other one, does not harm the one that you fulfilled. If you established your salah, you fulfilled a commandment for him. Walking around without using a veil, doesn’t prevent you from performing salah. Also it does not break or harm your wudu.