This is the view of different maraje regarding permanent marriage with People of the Book:

Imam Khomeini: As an obligatory precaution, it is impermissible to get married to People of the Book.

Ayatullahs Golpaygani and Safi: The permissibility of marriage with People of the Book isn't devoid of strength [this is a jurisprudential term meaning that it is permissible], but in the case of one being able to marry a Muslim woman, it is greatly makruh, as a matter of fact, in such a case, precaution should be observed. [1]

Ayatullahs Khu’i and Tabrizi: As a recommended precaution one shouldn’t permanently get married to them and some sects such as the Khawarij, Ghulat and Nawasib who consider themselves Muslims have the same ruling as kafirs and can't be married to by Muslim women and men, neither permanently nor temporarily.

Ayatullah Nouri Hamedani: Marrying them permanently is permissible according to the stronger fatwa, but precaution should be observed in not getting married to them permanently in the case of them being able to get married to a Muslim woman.

Ayatullah Fazel Lankarani: The Muslim man cannot be permanently married to kafir women, even if they are Jewish or Christian as an obligatory precaution, nevertheless temporary marriage is permissible with Jewish and Christian women.

Ayatullah Makarem Shirazi: The Muslim man cannot get permanently married to kafir women as an obligatory precaution, but temporary marriage with People of the Book such as Christians and Jews is permissible.

Ayatullah Bahjat: The Muslim man cannot get married to other than People of the Book and Magi [from the kafirs], neither permanently, nor temporarily, and apparently, temporary marriage with Jewish and Christian women, both temporary and permanent, isn't void although it is makruh and against recommended precaution, especially in permanent marriage.

Ayatullah Sistani: The Muslim man cannot get married to kafir women who aren't of the People of the Book, but temporary marriage with Jewish and Christian women is permissible, and according to obligatory precaution, permanent marriage with them isn't. [2]

Considering these fatwas on marriage with People of the Book, if you follow one who sees permanent marriage with them as permissible, your marriage will be valid. Even if your wife was Muslim, her permission wasn’t necessary for temporary marriage, therefore in your case you definitely won't need to get permission. Of course, if you were married to a Muslim and wanted to get permanently (not temporarily) married to a non-Muslim, you would have to get her permission. [3]

For further information on this issue, please refer to the following:

The reason for the impermissibility of the marriage of Muslim women with men of the People of the Book, Question 2723 (website: 2980).

Temporary marriage with women of the People of the Book, Question 7779 (website: 7903) .

Temporary marriage of a married man without his wife’s consent, Question 708, (website: 749) .