You would have to move every electron traveling in between each neuron into exactly same place, same momentum etc. All of them in the same time obviously. I stress the word move. In a system that is constantly moving on molecular level at crazy speed. Even that is only valid in case our consciousness is just electron cloud at given moment travelling in the brain. Maybe its more, all interconnected.
How you want to move that. The science for that won't be around for millennia, if ever (there may be quite hard constraints on quantum level in similar way speed of light is hard limit for Star trek utopias, universe doesn't rearange itself and change its laws just because of our emotional wishful thinking). Once we can do that, it means we could mess with telomeres, DNA etc with 100% precision and knowledge so this will be a moot point. Don't hold your breath for this.
This is still within the science fiction territory, but much more realistic than fixing aging or an actual teleporter.
The brain would still be old, but it would probably benefit quite a lot from the new body's youthness.
Bacteria do that in ~hourly intervals, humans take decades, some plants can span millenia.
So you could say the 'update interval' is tuned to how fast-changing a species' environment is. A balance between energy 'wasted' on re-building individuals from scratch, vs wasting energy by having poorly adapted individuals.
So 'immortal humans' to me reads as: humans optimized for a caveman hunter-gatherer lifestyle, while living in Star Trek like tech-heavy surroundings.
If anything, humanity would be served by shorter lifecycles, until tech advances stabilize somehow (?). Okay a few individuals living way longer could be good. Eg. billionaire tech bros taking that spot? Please nooo!!!