Harvest now, decrypt later
Recording encrypted traffic today in order to break it in ten years — the reason migrating to quantum-safe cryptography is urgent already.
The attacker needs no quantum computer to begin. Storage is cheap, and an actor able to tap backbone links can archive traffic for decades. The day a sufficiently large quantum computer exists, the whole archive can be decrypted retroactively, because the key exchange in TLS rests on mathematics that Shor's algorithm breaks.
What determines how urgent this is for you is how long your data must stay secret. A session cookie does not matter. Medical records, source protection, state secrets and engineering designs have lifetimes that overlap comfortably with when the machines are expected. That is why the large operators began rolling out hybrid key exchange in TLS well before the threat became acute.