All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
First of all, we thank and praise Allaah Who guided you to Islam and we ask Him to make you firm and steadfast on His Religion until the Day you meet Him.
Dear sister, we advise you to seek beneficial knowledge, perform good deeds and befriend righteous and pious women, as these are the best things that help you remain steadfast on your religion.
As regards your question, then if a woman embraces Islam and her husband is still a non-Muslim, then her marriage contract with him becomes void. In case he embraces Islam while she is still in her waiting period, then she is still considered his wife by the first marriage contract, but if her waiting period ends before he embraces Islam, then it becomes permissible for her to marry someone else as we clarified in Fataawa 107894 and 84226.
Therefore, if it becomes evident to you that your husband did not embrace Islam until your waiting period expired, then you are Islamically permitted to marry someone else as you are no longer his wife. According to the man-made laws, you are still considered the wife of the first husband until divorce or official separation takes place. In any case, we advise you not to marry until you legally separate from the first husband otherwise this may cause you harm and lead to bad consequences.
With regard to finalizing the procedures of divorce and the like, like the period that the procedures of divorce will last according to the law, then we are not aware of it. However, you should take the initiative to do so and you may know the procedures that you should take by asking the people who are experienced in this field, and you may seek the help of some trustworthy Muslim brothers in this regard.
Concerning changing your name in official papers, then this is not obligatory, but it is not permissible for you to be named after the name of your husband as we clarified in Fataawa 118807 and 89932. Also, you are not permitted to change your daughter’s name to any other name than her father’s name as this is Islamically forbidden as we clarified in Fatwa 154949.
In any case, a Muslim woman should endeavor to live a Muslim life especially in regard to her morals and dealings with others so that she would show to the people the good traits of Islam and encourage them to embrace it.
As regards buying a house, then if you fear you will be harmed by buying it on your name, then you have two choices: either you wait until you become separated from your ex-husband, or that you buy this house on the name of another trustworthy person while writing a legal document that this house belongs to you.
Allaah Knows best.