How long does it take for an edit to propagate from OSM to Mapbox's tiles? Average/worst case. I know this is a tough problem, so I don't envy you. :-)
Well, they probably have some kind of tile "lint", need to precompute a bunch of routing data (depending on what their actual algorithm is), sync all their rendering/prerendering pipelines at a common revision, propagate outputs to cache servers, etc. So it can't easily be done at every change, unless they have a design where small incremental deltas are explicitly supported by all their services. It is a time- and resource-intensive process. Still, three months sounds like a lot.
The concern seems to be moderation (but they haven't made a clear statement about it either). The current pace of updates is slower than what they were doing a year ago, so it probably isn't a technical thing.
Their routing engine updates more frequently than the rendering data.
OSM sub-10 minute updates really depends on zoom level and data. With low zooms the update can take weeks even there, as I recently learned in a hard way. It is more important to have consistent and laundered map data than someone's last minute changes.
What kind of places details are you looking for?
Depending on the geographic region, Mapbox has data for regions, cities, addresses, zipcodes, and points-of-interest (POI).