Quantom computers are going to be much "slower" in terms of becoming a problem. We just had the public disclosure of Meltdown and Spectre, and I suspect that, with what is known publicly, a non trivial amount of CS undergrads would be able to successfully craft an exploit based on this.
In contrast, even if we suddenly solved the engineering challanges of building a large quantum computer, it would likely take a while for the economics to work out where a quantum based attack would be economical. For one, for the foreseable future, quantom computers would be expensive (needing high cooling at a minimum), so a 'casual' attacker would need to wait for an economical rental type service to emerge. Also, it would take time to scale up production, so the first bunch of quantum computers will be expensive due to market pressures from big players.
In reality, it is likely that quantom computers would see a gradual (even if exponential) rise in power that would give some idea of a timeline for when we have to adapt. Not to mention the research into replacing quatom vulnerable crypto with quantom resistent crypto for over a decade, and has already led to (seemingly) quantom resistent replacements.