Is it permissible for a woman to pray in the mosque with men in the last row, as was the practice during the time of the Prophet, peace and blessings be upon him, or is the segregation of women from men an innovation?

You were correct in performing the prayer on time due to the absence of a prayer room, for the earth is a mosque and a means of purification. It is permissible for a woman to pray in the mosque behind men if there is no fear of temptation. Indeed, women used to pray behind men in the mosque of the Messenger of Allah, peace and blessings be upon him. Separating women from men in a dedicated prayer area is not considered an innovation (bid'ah), because Islamic law indicates the permissibility of taking precautions to prevent mixing.