Users who register at wikitech automatically request shell access rights for Wikimedia Labs (cf. https://wikitech.wikimedia.org/w/index.php?title=Special:Ask&q=[[Category%3AShell+Access+Requests]][[Is+Completed%3A%3Afalse]]&p=format%3Dtemplate%2Flink%3Dall%2Fheaders%3Dshow%2Fsearchlabel%3DOutstanding-20Requests%2Fdefault%3D%28no-20outstanding-20requests%29%2Ftemplate%3DShell-20Request%2Fintrotemplate%3DShell-20Request-2Fhead%2Foutrotemplate%3DShell-20Request-2Ffoot&po=%3FShell+Request+User+Name%0A%3FShell+Justification%0A%3FModification+date%0A&sort=Modification+date&order=asc&eq=no). These are then processed by a wikitech user of the "shellmanagers" group.
It would be nice to graph how long users have to wait for their request to be processed.
The simplest query would be to go to the above link, click "Edit query", remove anything but "?Modification date" from "Additional data to display", choose "JSON export" (or whatever) as "Format as:", set "limit:" to 1 and click "Find results".
This should produce (a repeatable link to) a JSON (or whatever) file. If there are no records in the results, the processing time is 0, otherwise the difference between time() and the "Modification date" printout.
For historical data (which I don't think is needed at the moment), this gets more complicated as you would have to iterate over all pages in the shell requests category that are not open requests, if there are only two revisions in the page history, the processing time is the difference in the revision timestamps, otherwise you would have to go forwards in the revision history from the second revision on and parse the wikitext to see if the parameter "Is Completed" is set positively until you find the first one.