Using OSMConvert, the wiki shows how you can extract/display the timestamp from the file header:
osmconvert yourfile.pbf --out-timestamp
This could be included in your update process and piped into the database for your GUI to read later.
you can download it here
Using OSM2PGL, the wiki describes how the timestamp is retrieved with each object by editing the default.style file (or create a new one). A new column is created in the database for the extract. Simply unremark the timestamp at line 78 and use the following command without --hstore and include the style file reference:
osm2pgsql --extra-attributes -S default.style -d databasename yourfile.pbf
You don't have to get that from the database. When you are downloading the PBF you can retrieve the timestamp at that time.
If you are using wget then you can use the -S option (server response) which downloads and produces this result containing the last modified date:
or access the file system in the programming language of your GUI to get the last modified date:

