Thank you for sharing with us your problem dear brother, a problem that you are not alone in.

There are thousands of Muslim doctors around the world which indicates that some solution must have been found, between their acts of worship, and their o take work as a form of worship. To take care of the sick is an act of worship, for worship in Islam is not simply praying five times a day, but is in our deen (our religious life transaction). The problem occurs when we separate our prayers from everything else that we do.

It might seem as if the Director is being rather harsh or even anti-Muslims, but in fact he is being very realistic by presenting to you the reality of the career which you are embarking on. There may be indeed no real conflict between both acts of worship, and the fear you are experiencing is just presenting obstacles. As much as it is preferable to pray on time, it is allowable in Islam to pray within time with the exception of Salat-ul Fajr (dawn), and Salat-ul Maghrib (dusk). It is even permissible to pray two prayers together i.e. Salat-ul Dhur (early afternoon) with Salat-ul ‘Asr (early evening), and Salat-ul Maghrib with ‘Ishaa (late evening).

"The time of the noon prayer (lasts) as long as it is not afternoon, and the time of the afternoon prayer (lasts) as long as the sun does not turn pale and the time of the evening prayer (lasts) as long as the spreading appearance of the redness above the horizon after sunset does not sink down, and the, time of the night prayer (lasts) by midnight and the time of the morning prayer (lasts) as long as the sun dots not rise" (Muslim 4: 1273).

"Ibn 'Abbas reported: The Messenger of Allah (may peace be upon him) observed the noon and afternoon prayers together in Medina without being in a state of fear or in a state of journey. (Abu Zubair said: I asked Sa'id [one of the narrators] why he did that. He said: I asked Ibn 'Abbas as you have asked me, and he replied that he [the Holy Prophet] wanted that no one among his Ummah should be put to [unnecessary] hardship.)" (Muslim 100: 4 #1516)

However, remember that it only takes 5 minutes to pray! As you grow accustomed to the pattern of work, and the layout of the building, you will note when it is possible to take 5 minutes to pray insha-Allah.

For further guidance, please try the following link(s):
Observing Moderation in Worship
Taking Time Out With Allah