What is the ruling of Islamic law regarding a husband remaining with his Western wife who embraced Islam for the sake of marriage, yet shows no signs of commitment to the religion, and does this affect the upbringing of their son?

A Muslim woman, by uttering the two testimonies (Shahadatain) and then falling short in her actions, is not cast out of Islam. She should continue to be advised, and it is permissible to separate from her. We should note that a disbeliever (kafir) does not truly enter Islam except by uttering the two testimonies with conviction, for the Prophet, peace and blessings be upon him, said: "No one testifies that there is no god but Allah and that Muhammad is the Messenger of Allah, truthfully from his heart, except that Allah will forbid him from the Hellfire." And knowing this is difficult. If an original disbeliever utters the two testimonies and then says: "I did not intend Islam," he is not to be believed and is considered an apostate.