I had the opposite experience. Uber has customer support built into the App. I was also billed twice and just followed the instructions in App that created a ticket. Got my money back in 2 days.
Same thing in my case. I had a few mishaps - one guy didn't terminate the trip once I got off, so I got overcharged, another failed to pick me up to begin with (!), but started a trip anyway, etc. In every case the issue was financially resolved in terms of hours.