Winter boots are best.
The troubles with trying to use your normal shoes are:-
a) they will probably have pretty open weave material above the toe that let the draught in and the warm air out.
they are generally a close fit to suit thin summer socks. If you try and cram thick socks in you just end up restricting the circulation to your feet and making bad worse.
c) stiff soles and the consequent lack of movement of the foot doesn't help.
The best you can do is to use medium thickness wool socks and neoprene overshoes, and to wear your shoes relatively loosely.
Walking for a bit to get the blood flowing does make quite a lot of difference if things get too bad.
One time when I was working in the Antarctic, two of us skied the 10 miles back to base in about -30. I was using XC skis, with flexible boots that were just a single layer of leather, the other chap used mountaineering skis and rigid double plastic boots. I was fine, he got frostbite to the tune of his big toe turning black.