Maybe it'll require TLS 1.4/QUIC 2, with no changes but the cipher specifications, but it can happen in two or three years. Certificates themselves don't last longer than a year anyway. Corporations running ancient software that doesn't support PQ TLS will have the same configuration options to ignore the security warnings already present for TLS 1.0/plain HTTP connections.
The biggest problem I can imagine is devices talking to the internet no longer receiving firmware updates. If the web host switches protocols, the old clients will start dying off en masses.