i'm in a similar situation to you.
had a small (1g) tank with ramshorns in it. during the summer it was on a windowsill, was always warm and was overrun with algae (on purpose i might add).
i didn't purposely cycle or even filter the tank, just had an airpump / stone running on it.
they bred like, well - snails. hundreds of eggs + baby snails. the water was filthy due to overfeeding, no filtration and snail poop. out of curiosity i tested the water once and got ammonia = 0, nitrite = 0, nitrate > 50ppm. seems the tank was cycled.
anyway, i moved house and set the snail tank up in a cooler room with no strong light source. also the weather got cooler. the result was no more snail eggs. so i've just bought a 10g mini tank setup with filter, heater, lights and have planted it. I seeded the filter + gravel with stuff from my old tank and added some harlequin rasboras and the remaining large snails.
the tank never tested positive for ammonia or nitrite and i'm not getting much of a nitrate reading due to the plants, but it is fully cycled.
but the snails still haven't started laying eggs again!!!
not sure if any of that answers your question mind! i would say that when you add snails to a tank, it will cycle naturally due to the ammonia produced. the snails will not die (IME) as a result of this.
HTH