the partnership situation is basically waymo realizing they've grown enough on their own that they don't need uber as much anymore. waymo one (their independent service) has been expanding and making solid money, so there's less reason to split revenue with uber when they can keep everything themselves. if the partnership ends, you might see waymo rides disappear from the uber app in cities where waymo operates separately, but that doesn't mean you lose access to autonomous rides - you'd just book them through the waymo app instead.
for most uber riders this honestly won't change much day-to-day. waymo only covers specific zones in a handful of cities like san francisco and phoenix, so unless you're regularly in those areas ordering autonomous rides through uber, you probaby won't notice anything different. regular uber with human drivers keeps running either way. the bigger ripple is for people who specifically want driverless options in those cities - they'd need to switch apps rather than having everything in one place.
here's something that might help if you're in a waymo service area: check both apps before a trip. sometimes waymo pricing and uber pricing differ for the same route, and occasionally waymo has promotions that uber doesn't advertise heavily. i've found that opening both apps takes like 30 seconds and can save a solid amount depending on demand. also if you're curious about autonomous vehicles but hesitant, waymo's pretty transparent about where their cars operate - you can check their coverage map so you know ahead of time whether you'll get a driverless option in your area.