I used to cycle through snow and ice to work wearing waterproof socks with an extra inner pair of socks (Sealskins). Worked well for warmth. Only 45 minutes though. Realistically, longer rides your feet will always sweat though and none of these gore-tex type materials wick away water quick enough.
Wet doesn’t matter really if your feet stay warm though (single day: don’t want to risk trench foot…).
Are your cycling shoes big enough for 2 pairs of socks? I deliberately bought a pair big enough for two. If you can wear a normal thermal pair and then over them some waterproof ones.
[pick up some free newspaper now, to stuff in your shoes on your return. Putting on chilly damp shoes is grim…]