For a 22uF capacitor, and where the time required is 15 seconds, the series resistance is about 620k.
The trigger point for Vcc of 15V is about 10V, at which time there will be 5V across that resistor, and so the current through it will be about 8uA. According to the 555 specs, you may see 0.35uA leakage through pins 6 and 7 and about 3uA through a typical electrolytic capacitor.
This means that almost half the current used to charge the capacitor is leaking away. Leakage tends to be quite temperature dependant, so this figure will not be constant. If you see temperature variations, you will see timing fluctuations in excess of what you might expect.
You are in the region where the normal calculations for resistor/capacitor values for the 555 do not apply, but probably not yet close to where the device simply fails to ever get to discharge. The timings are not easily predictable, but are probably reasonably repeatable.
Practically, because you're almost certainly going to set these resistors by observation (you've considered this, right?) -- the initial approx 2.5% inaccuracy of the 555 (which incidentally is only for R <= 100k) is probably OK.
Again, from memory, I think you specified 2% accuracy (on checking, I note that you say 5% is acceptable).
I would not be comfortable with expecting 2%, but 5% is achievable. It does depend somewhat on the capacitor -- I would be looking at using a tantalum device.
However a major issue is that the timer will spend a lot of its time with the capacitor voltage near the trigger point. During this time it will be sensitive to noise. I would recommend that you ensure that your power supply is quiet and that the 555s' power supply pins are bypassed.
What is the impact of one of the 15 second monostables triggering early?
Would I use 555's for this? No.
Would I advise you to use them? No.
It looks like you've latched on to a particular idea and want to make it work when there may be better and simpler solutions that do not suffer from the drawbacks that this method does.
This particular method also requires a button press to start it. Is that a requirement? There is no way to stop it (other than powering off) -- is that what you want? Maybe those are un-stated requirements and possibly tend to favour one approach over another.
One of your responses in another forum suggested how simple a picaxe program would be to solve this (and personally I think it's probably the simplest option). Had you come back to us with the question of how to drive a relay from a picaxe, we could have answered very quickly (it's one resistor, a transistor, and a diode).
I note that we are your "friends" in other forums, and they are your "experts" here. Why do you feel the need to keep your other sources for assistance anonymous? Nobody here is going to get upset if you say "the people at xxxxx say...".
You got some very good advice from allaboutcircuits.com, such as the need to have a resistor in series with the pots (I assume trimpots), and issues concerning reset at power on. Incidentally the issue of reset at power on is something that could cause you lots of grief if 2 or more of the monostables decide that they're triggered at power-on. You could end up with 2 signals chasing each other around the loop, rather than the single one you want.
