When arranging travel I find myself google 'does X have Uber?' a lot (substituting a different city for 'X' each time).
Is there a complete, up-to-date list somewhere?
Travel Stack Exchange is a question and answer site for road warriors and seasoned travelers. It only takes a minute to sign up.Sign up to join this community
A search on Google would have found you this page. 700 cities listed on the Uber web site.
@Arthur has provided a good base to answer your question. And to be fair, to quickly check if Uber is present in a specific country - it's probably the best and easiest way.
However, for identifying whether Uber is present in a specific location i.e. city, town or a village, I will still advise to actually open the Uber app to see if there are any Uber cars in that location. If the location is not officially in the list that doesn't mean you won't find Uber cars there!
Alternatively to using the Uber app, it seems entering your location on UberEstimator provides the same information, not sure how accurate it is but it gave positive result for Milton Keynes in the UK, which is not on the Uber website but is indeed on the Uber app and on UberEstimator website: