This is difficult to say, and (in my opinion) largely depends on the price of self-driving cars.

If Self-driving cars become available but are unaffordable (due to very expensive equipment onboard), Uber will be in an excellent position to make an investment in these cars where the average taxpayer will not.

Uber will then be rid of it's largest cost, human drivers. They will have the dispatch system in place to organize their self-driving vehicles and will be able to offer their service at an incredibly low price compared to the price of buying a self-driving car. Uber looks at a massive upside.

If self-driving cars are offered affordably, Uber can still offer it's services way cheaper and nothing changes on the consumer side. People still have cars. Theoretically, the demand for their service would still increase.