I don't think brain uploading would work in the sense of “you upload your mind to a computer”.
The process might look something like this:
Importantly, you don't simply wake up and experience life as this newly created entity. There is a loss of "cognitive continuity".
What might work is gradually replacing brain cells or body cells with artificial ones that function identically, but do not age or die. This could potentially lead to what could effectively be called immortality, though it's still possible to destroy this entity. This idea presents a bit of a ship of Theseus paradox, but it could preserve the continuity of the "software" while replacing the "hardware". Perhaps.
There are still issues with this approach, since the difference between a standard biological system and an effectively immortal, artificial one is still massive (hint: it's the "immortal" part). It raises the question, with such a disparity, would life still be comparable to the human experience? The presence of these “low-level” changes often cause emergent issues in higher order functions.
Also, what makes a cell immortal, or what causes the body to age? Uncontrolled proliferation of cells is essentially cancer, and aging is caused by DNA telomere shortening. It seems more realistic to address the latter scenario. Then, if we introduce “non-shortening artificial DNA telomeres in an otherwise identical artificial cell”, it suggests that we likely discovered how to implement this in biological cells. At that point, why bother with the artificial replacement?
There is another thing to consider: if we made humans "effectively immortal" by eliminating aging and disease while retaining the physical body, sooner or later everyone would likely meet a rather violent end. I also have doubts about the very proposal that immortality or extremely extended lifespan would be something we would enjoy at all. Of course, there is the possibility of an extended lifespan and healthspan, but it doesn't rule out people growing weary of life. If you live for 300 years or so and witness 300 years of wars and whatever other human evils, it would certainly take a psychological toll on you. Physical "improvement" does not necessarily translate to moral improvement.
On a somewhat related note: Living in a matrix-like simulation, where you retain your original body or brain but your experiences are simulated, would likely cause severe psychological issues sooner or later. Simulating every little aspect of real-life experience is incredibly challenging, and humans are great at detecting the uncanny valley (consider how a video game that seemed extremely realistic a few years ago now appears flawed). Physical feedback and sensory experience is still essential for things to feel, well, real. Not having them would probably result in a mass epidemic of depersonalisation / derealisation related disorders, and probably the emergence of a couple psychological issues that don't even exist yet.