Is it permissible for me to wake my wife for Fajr prayer knowing that she breastfeeds her small daughter during the night and sometimes she is very exhausted?.
Prayer is very important, and we are commanded to offer prayers at the prescribed times; we are urged to do that and we are warned against neglecting it. This is well known. For example, Allaah says (interpretation of the meaning):
“Verily, As‑Salaah (the prayer) is enjoined on the believers at fixed hours”
“Guard strictly (five obligatory) As‑Salawaat (the prayers) especially the middle Salaah (i.e. the best prayer ‑ ‘Asr). And stand before Allaah with obedience”
And the Prophet (peace and blessings of Allaah be upon him) said, when he was asked which deed is most beloved to Allaah: “Prayer offered on time.” Narrated by al-Bukhaari (527) and Muslim (85).
So you should beware of missing prayer or delaying it until the time for it is over, because Allaah says (interpretation of the meaning):
“Then, there has succeeded them a posterity who have given up As‑Salaah (the prayers) [i.e. made their Salaah (prayers) to be lost, either by not offering them or by not offering them perfectly or by not offering them in their proper fixed times] and have followed lusts. So they will be thrown in Hell”
It is not permissible to delay prayers beyond their appointed time unless that is for a reason that makes it permissible to join prayers at the time of the earlier or later prayer. It is well known that Fajr prayer cannot be joined with any prayer before or after it, so it must be offered on time, no matter how sick or tired a person is, so long as he is conscious and of sound mind. He should pray standing, but if he cannot then he may offer it sitting, and if he cannot do that then he may pray lying down.
The sleeper should take precautions to help him wake up, such as not staying up late, setting an alarm clock and asking others to help him wake up.
The husband is enjoined to take care of his family and save them from the Fire, as Allaah says (interpretation of the meaning):
“O you who believe! Ward off yourselves and your families against a Fire (Hell) whose fuel is men and stones, over which are (appointed) angels stern (and) severe, who disobey not, (from executing) the Commands they receive from Allaah, but do that which they are commanded”
Based on this, you should be keen to wake up your wife and encourage her and help her to pray; you can delay waking her up until it is nearly the end of the time for prayer. We ask Allaah to guide and help us and you.
And Allaah knows best.