So you refer to the timestamp that a osm object had been touched (or created)?
I would recommend to work offline and get an appropriate Planet extract. Sadly AFAIK there is no tool that works with OSM that way, as they use a slightly different method for replication (based on changesets and version numbering).
So you need to do it manually:
- download .pbf extract that you need
- cut it using osmosis till it fits your area and export it as .XML
- create custom script that filters OSM XML basing on the timestamp="" XML attribute.
Due to the amount of data, you need to use a SAX XML parser instead of DOM. You might also consider, that to reconstruct a datasnapshot, nodes can be changed independently to ways and both of them interdependently to relations.
I don't know your exact usecase, but maybe the history planet is a better way to get your results. This depends on the size of your area and what you like to get. If you just like to view the date of your area, you might use OSMatrix and to monitor, you might have a look at this tools: