Someone prays all Salat on time except the Fajr prayer which he misses for many reasons. One is that he watches TV, etc. at night and fails to get up in time to pray Salat Al-Fajr. Will Allah (SWT) accept from him the four prayers if he doesn't pray Salat Al-Fajr?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

It is not permissible for you to make a habit of praying after the fixed prescribed time of the Prayer has elapsed. Allah Says (what means): {Indeed, prayer has been decreed upon the believers a decree of specified times.} [Quran 4:103]

If you are neglectful and fail to give importance to arise for Prayer, then you commit a sin. So, you should take all means that help you to awake early such as using an alarm clock or watch, or asking someone to wake you up, or other means.
You are obliged to get rid of this dangerous bad habit. Allah Says (what means): {But there came after them successors who neglected prayer and pursued desires; so they are going to meet evil.} [Quran 19:59]

Ibn ‘Abbas said that they neglected the Prayer, either by not offering it, or by not offering it perfectly, or by not offering it at its proper fixed times, etc.

Anyway, you have to pray the Fajr Prayer even after its time has passed if you had not performed it on time. Only Allah knows whether it is accepted or not; however, by doing so, at least you fulfill your obligation.

If missing the Fajr Prayer is due to staying awake at night, then you are not permitted to stay awake late at night.

Moreover, staying awake at night becomes worse if you pass your time in watching T.V programs that are full of shameless and forbidden acts. As you know, many T.V programs contain music, lustful songs, naked women, repeated news, and useless programs which are not permissible in the Sharee’ah.

For more benefit on the ruling on abandoning the prayer or delaying it, please refer to Fataawa 384457, 377344, 88016, 84178, 83565, 87587, 90926, 88690 and 84403.

Allah knows best.