Do what makes sense for the story
FAE makes it very clear that while the durations are solid suggestions for what makes sense, it's the "making sense" part that's more important than their suggestion.
- Moderate consequence: Clear it at the end of the next session, provided it makes sense within the story.
- Severe consequence: Clear it at the end of the scenario, provided it makes sense within the story. (FAE 24)
To me, this means that at the start of the next scenario a severe consequence could maybe be assumed to have received the proper attention, but is still being dealt with. Since the idea is that "moderate and severe consequences stick around for a while" (FAE 24 sidebar), you should ensure that they do; otherwise it wouldn't make story sense.
A consequence should have the opportunity to make an appropriate impact on gameplay, and that means sticking around for an appropriate amount of time.
If I were to postulate a rule of thumb, I'd say that a severe consequence should last at least one session longer than a moderate consequence, or at least 1/4 of the time it takes to tell a scenario (whether or not that time spans just one scenario or spills over into the next), whichever is longer.
For a more concrete bookkeeping rule, try this: In addition to needing an explanation for recovery, a consequence cannot be removed until it has been successfully compelled or invoked against (granting the player Fate points) a number of times equal to the amount of stress it could absorb. This removes the normal duration requirements for recovery, except that consequences greater than mild must last at least one session.
There's another option you may not find appealing, but I'm going to mention it for completeness's sake: drop severe consequence slots entirely. If you're running one-shot scenarios, maybe mild and moderate consequences are sufficient. Of course this would change the nature of combat a bit, but it might be interesting! I'd love to hear how that works out, if you try it.