I have difficulty imagining any way Uber can compete with an entrenched last mile provider on cost. What possible economic advantage would individuals have in parcel delivery versus an entrenched provider? UPS has massive benefits from economies of scale in terms of route density / fleet maintenance / fuel costs / etc etc, which no individual vehicle operator will ever be able to compete with.
Sure, an Uber driver could charge less for labor, but given that the cost of operating their vehicle will inevitably be more expensive per delivery I expect there is probably no price point at which they can realistically compete.
Perhaps Uber drivers have a path forward in niche delivery services: food, very time sensitive deliveries, rerouted deliveries, etc. But in those areas they will need to compete on quality, and the market for more expensive, higher quality deliveries (while possibly extant) cannot be large enough to represent an existential threat to a company the size as UPS.
Beyond this, I'm curious as to whether it's even in UPS's best interest to continue to service the last mile.
Both UPS[0] and FedEx[1] already have services wherein they partner with the USPS specifically to avoid servicing the last mile. Outsourcing to better/cheaper/more efficient last mile service providers might eliminate a lot of UPS employees and revenue, but the last mile is incredibly expensive to operate. Is the last mile service really a major source of profits when compared to their highly efficient long distance operations?
[0] http://www.ups.com/content/us/en/resources/track/sp_definiti...
[1] http://www.fedex.com/us/smart-post/index.html