Not personal experience, but just a funny (thread topic related) anecdote...

Went into the NEXUS enrollment center at YYZ (got a new work permit and needed it filed) and someone signed in at the update book (they have a sheet for appointments and a sheet for updates like mine) asking if they could get the application number, having applied two months ago. The CBSA officer seemed shocked that someone in their twenties would apply by mail and said that until they received some sort of reference number (E.g. PASS ID) there is nothing they could do. Commented (not to the person's face) that people should apply online and "catch up with the twenty first century".

So it definitely seems as though paper applications are significantly slower...

Anyhow, on an semi-related note, apparently if you have almost doc expire (work permit, visa, etc.) then it's an automatic flag on NEXUS for IMM 1-1. Until you visit an enrollment center, it won't get cleared.