How fast do tantalum capacitors self-heal?
I guess I'm confused by your question. If I'm reading your question right, you've seen the capacitors fail, and you're wondering why they failed and how long it will take them to self-heal.

To answer the "why" of this situation: If they failed, it's not because of an overvoltage. Yes, they're not derated enough, but they should still withstand 15 VDC without failing immediately. It's most likely because your 15 VDC isn't current-limited, and the current surge triggered a short. This is a common and dangerous failure mode of Tantalums.

As far as self-healing, in general, self-healing should happen rather fast. But to be clear it will only happen during operation since it's based on the localized heating due to current passing through the dielectric fault in the capacitor. If you're just leaving a tant unpowered then it will never heal.

Also, you can only rely on self-heating for fixing mild shorts/dielectric defects at reasonable currents. If you put an non-current-limited 15 VDC across a Tant with no slow-start circuit then you will get a shorting failure mode from the current surge and something will explode long before the cap self-heals. Self-healing should be relied on to fix manufacturing defects in the dielectric, not full-on shorts due to a current surge.