Whether we think of taxi services or privately owned vehicles, I don't see Tesla, Cruise/GM, Intel/Mobileye, or indeed any other venture which is running actual rides with no one in the driver's seat. Not to mention the disengagement data from California, which Waymo was doing very well on. Granted, overall, all of these teams are being secretive, so it's tough to know where they truly stand. Still, based on what we've seen, I say Waymo is out in front by at least 6-12 months.
When it comes to delivering SDC to the entire globe, obviously there will be a geo-spatial distribution of roll-outs, with sunny/dry/well planned out places (i.e. Arizona, California) getting SDC first, and snowy/wet/poorly organized (i.e. Boston's cowpaths) cities getting them much later. Not to mention, there need to be lax regulations that remove obstacles to local adoption, and super long term we'll see SDC-specific infrastructure improvements to utilize their inherent advantages/disadvantages.
But, for now, it seems to me that Waymo is in the lead and that the SW USA and California will get all of these Waymo tech in Chrysler/other vehicles first.