updated 09:39 am EDT, Tue August 5, 2014
Phones must be examined before replacements are authorized
(Updated with European news) As predicted, Apple Stores have officially begun replacing iPhone 5s screens. Accounts indicate that the procedure can take as little as an hour, and is rolling out across the US, although it may not be available at every location. Previously, if an Apple Store didn't replace a phone outright it would ship the unit out for service, something that could take several days.
An on-site screen replacement typically costs $149, but that amount ultimately depends on how the screen was broken and whether or not someone has AppleCare+, which can lower the fee. In any case, accidental screen breakage isn't covered by an iPhone's default one-year warranty. An iPhone may still end up being replaced or sent off-site if examination shows more substantial problems.
It's not clear what's been responsible for the delay in offering 5s support, since 5c screens have been swapped out since January. It's most likely an issue of part supplies, since the 5s is significantly more popular and it may have been difficult to produce enough spare screens.
Update: Sources are saying that the program will expand to Europe next week, possibly as soon as Monday, August 11th.