If any kind of proof about serious quantum computers comes to light, browsers can force most websites' hand by marking non-PQ ciphers as insecure.
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.
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.