Waymo is ahead on the technology. They were ready to begin public trials with their Koala cars in 2015, but the California DMV shut them down, and many of their key people jumped ship.
Waymo has since been proceeding a bit more cautiously, and their efforts at this point are oriented more towards smoothing out ride feel, testing under adverse weather conditions, and their early rider program in Phoenix.
Whether it's Waymo or GM to officially be the first to have an empty car driving on public roads, the other won't be far behind, though the victory will be largely symbolic. GM's cars still need a lot of polish, and Waymo needs better manufacturing partnerships.