Is the prayer of one who woke up for work, remembered that the time for Fajr had come, and prayed it, acceptable or not? And what is the evidence for its unacceptability if that is the case?

The prayer mentioned is valid as long as it fulfills the conditions and pillars [of prayer]. Its validity is not affected by the fact that the worshipper woke up for work and did not bring to mind the intention for prayer, for the intention for work does not invalidate the intention for prayer. Its acceptance is with Allah, and a Muslim should bring to mind the intention of sincerity in all their acts of obedience. Imam An-Nawawi said: "It is befitting for anyone who intends any act of obedience—no matter how small—to have the intention present, which is to intend by their action the pleasure of Allah, the Mighty and Majestic, and their intention should be present at the time of the action."