If the Stormclouds of Doom (TM) really were looming that large on the hapless email-writer's horizon, they would either have sent more, increasingly meeping mail, or phoned someone.

I always clear backlogs from the most recent backwards - because (a) the person who's been waiting a month already thinks you're inefficient, another day or two really won't make a difference and (b) If it was really important, they sent another, more recent, more urgent mail to which you will then respond soon, and
(c) The recent ones then get super-efficient service, which outweighs the not-quite-as-efficient-service as the ones waiting a month got.
(d) By the time you get to the mails you would feel REALLY guilty about, you're all buoyed up with a feeling of efficiency.