Answer
Wa `alaykum as-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear sister, thank you for your question and for your commitment to your religion.
First, we implore Allah Almighty to guide your husband to the right path and grant you ease in all your matters.
As for your husband, you should exert some effort trying to know what doubts he has about Islam. If he refuses Islam and insists on returning to his religion, then you should be separated from him. It is not allowed for a Muslim woman to marry a non-Muslim man.
Responding to your question, Dr. Muhammad Salama, PhD in Islamic Studies in English and an Assistant Professor at the Faculty of Islamic Science, Al-Madinah International University, states,
It seems that your husband is hesitant and perhaps he has problems or misconceptions concerning Islam. Therefore, you need to approach your husband kindly and admonish him. Try to know why he changed his mind concerning Islam. You may seek the help of scholars and sheiks in your area to talk to him and remove any doubts he has about Islam.
If he insists on refusing Islam, he becomes an apostate and you have to be separated from him. The Qur’an states, “And if you find them to be believing women, then do not return them to the disbelievers. They are not lawful [wives] for them. Nor are [the disbelievers] lawful [husbands] for them.” (Al-Mumtahanah 60:10)
Then if he returns to Islam within your `iddah (waiting period), you may resume your marital life. After the elapse of the `iddah, you may marry another one or wait for him to return to Islam to return to wedlock.
May Allah help you overcome this situation and guide your husband to the truth, amen.
Allah Almighty knows best.