I wouldn't expect there to be a single cure that would change things "overnight". Even de Grey talks about 7 different categories of aging damage, each of which will need a different type of treatment; and those are just general categories, most likely there will be different treatments for different systems in the body as well. And he's probably somewhat optimistic in his description of the problem.

However, I think it's entirely possible that we'll make enough progress in enough different areas to reach longevity escape velocity in our lifetimes. It's not going to be a single breakthrough that happens overnight though.