The "common cold" or "flu" isn't actually just one virus, its a group of them that have the same/similar symptoms and treatment. Most strains of those are not contagious to other mammals, let alone reptiles, but some are. Better to play it safe and interact minimally while you are sick.
You wouldn't be able to easily distinguish between rhinovirus and particular strains of coronavirus that are transmissible to them without getting labwork. Coronavirus is not a new thing, many strains have been going around as "common cold" for ages, and a decent number do cross species, including reptile and avian.
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