I think there's less to worry about re rust on reeds than there used to be.
I'm 66, and when I was a kid, our windows were regularly completely frosted up on the INSIDE in the mornings in winter.
No double glazing or central heating back then. It's hardly surprising that steel reeds got rusty.
But in today's conditions, so long as you avoid storage in a loft or cellar, or car boot, for extended periods, I doubt if reed will suffer much rust.
I would guess that playing a cold concertina wouldn't cause any bother, as any moisture would evaporate in seconds with the warm air whizzing about.
Makes you wonder about stainless steel though. Has anybody ever tried it as reeds? I'm guessing it must give poor tone, otherwise it would be ideal. I have one with silver nickle reeds which never corrode, but the tone is just the same as brass, and I'm guessing the durability is much the same. It has had one replacement with a brass reed, in approx 150 years, so not too bad. (I'm guessing the others are originals)