Depending on who you listen to, the "quantum threat" to bitcoin is either (a) purely FUD to be ignored, or (b) an inevitable truth that's coming with the when? as the only question, such as in five-ish or ten-ish years out (2030-2040).
I'm making my call right now. Here's how I see this playing out.
Bitcoiners will bicker and nothing will get done. This is the most certain thing in this post (because we're already doing it).
Quantum computing, and I dislike the name because it connotes the mysterious and conjures multiverse nonsense, will in fact come to fruition.
My guess is that quantum first appears in a lab, either a government agency or, my best guess, in a private company's lab. There might be rumors that ripple through the bitcoinverse when word leaks out, but nothing will be done. The bickering might get ramped up.
Then, the quantum breakthrough will be announced publicly. They might even say that, as an experiment, they were able to crack a bitcoin private key. The bickering will amplify for sure and the we-need-to-do-something crowd (the fix-it crowd) will gain momentum. But, the ossify-forever crowd will hold their ground as well and nothing will change.
Eventually, the secrets of the how-to and the hardware itself will leak out and there will be a real "hack". That is, a quantum computer will decipher a bitcoin private key somewhere beyond a lab and shuffle funds. It might be a small, unimportant bitcoin hack, but it will happen in the open.
The fix-it crowd will scramble to get their plan all but finalized and ready to roll out.
Meanwhile, there will be a big hack, maybe of an exchange. Maybe of some of Satoshi's coins.
This will be the breaking point -- certainly if Satoshi's coins, or just some of them, are compromised. Bitcoiners will call for an immediate "freeze" of the blockchain. (How that would work, I'm not sure! But, aligned incentives would be to halt the chain.)
Then, bitcoiners will call for a "rollback". Even if the hacked chain continued (which it likely would), a rollback would be called for. A "fix" of some sort will be worked out and the chain will be rolled back to a pre-hack block.
In short, (1) quantum computing will come, (2) the community will bicker, (3) quantum will hack bitcoin, (4) a fix will be developed, (5) then the chain will be rolled back and restarted.
I hope I'm wrong. I'd rather a plan get made prior and implemented, not just discussed. However, I'm doubtful that that will happen.
We'll see how this ages.
I go by 
