Is the questioner liable for the sin or expiation of his relative's death, especially since doctors mentioned that his delay in reaching the hospital was the cause of death, knowing that the questioner was unaware of his illness, and his father was not present at home at the time of the call?

You were not the cause of your uncle's death. A person is not held accountable for someone's death except through direct action or a pervasive cause. The deceased had fulfilled his lifespan. Faith in Allah's divine decree and destiny guides the heart and brings it peace after calamities. And whoever believes that a calamity is by Allah's permission, Allah guides his heart and grants him steadfastness and patience.