A small sign of that: I generally order Ubers from Google Maps. Until recently, I was sent to Uber app; lately, it’s an entire experience through GMaps. I suspect they are planning to switch that default to Waymo wherever they are.
I prefer GM because it has all my bookmarks, it doesn't download the same map tiles again, etc. Until recently, it was quite annoying, because you were trapped in "Uber mode" during a ride. As of a few months ago, you can go back to the base map, start new searches, look up transit directions, etc. and later return to your ride's status.
That does mean that Google Maps will win and its default option too. I suspect that the way they display services, notably prices and availability will have a dramatic impact too: at the moment, it’s a band. It’s all too easy to show a cheap low price for instance and I suspect that Google Maps might want to argue that they should review estimates and compare them to actual charges to enforce consistent labelling. Same for availability: you can say a car will be there in three minutes, but have that change to five once you clic, etc.
Overall, it does look like an anti-trust issue. I guess, in that case, a China-Wall between Google Maps and Waymo, transparent ranking process, etc. would be necessary. I can’t imagine not having access to either a mapping service, or the convenience of comparing public transport with ride-hailing.
I’ve been an advocate for auditing a lot of algorithms by publicly-appointed experts for a while now. You need a good understanding of machine learning to audit the political impacts of Facebook News Feed and the Call to Vote button, for instance. Not all countries can appoint their expert though, so we might need to have smaller countries agree to trust the expert from larger nation groups.
If you're asking whether you can ask maps to switch into the Uber app, I'm not sure there is a lot of value (you can pop open the app and your ride appears there too). Uber exposed developer APIs presumedly for this reason. Embedding the UX lowers friction and tightens up the user funnel.