Not entirely sure for a K2, but for my K6, I think the closest modern day equivalent is the Continental RB2 (front) and a K112 (ribbed rear). While they look period correct, there are probably better tires for back road carving.

I am changing them to Metzeler Lasertecs.
Another "retro" set is the Dunlop K70 if you want to go for that British look.
Avon Speedmaster (front) and Avon Universal 410 (rear) also has a vintage look.
There may be others with a retro pattern (Cheng Shin

), but the old school tires are getting harder to find.
Cheers!