Autumn planting seed potatoes are not specially bred variants of the ordinary types. They are exactly the same tubers that have been carefully stored.
The ideal time for planting is late August and plantings after the first week of September are unlikely to provide a Christmas crop. Most early and second early varieties are suitable potato varieties for late growing, but the easiest and most popular variety is Carlingford. I am growing Charlotte myself this year.

My grandfather stored his new potatoes in a sand filled biscuit tin buried in the garden for christmas.
He swore by King Edwards

King Edwards are a main crop potato. Your grandfather will not have grown potatoes to lift for christmas as it wasn't something that happened then. Duke of York is another popular one for grown to lift for Christmas. Personally like my roast potatoes with the turkey.

I think Mrbarrett's grandfather lifted the potatoes well before Christmas and then stored them as described. I have a gardening friend in his eighties who used to do the same in order to have some new potatoes for the Christmas table. He will be round here tomorrow and I'll ask him more about it. Thanks for the tip about using King Edwards Mrbarrett. I'll find out what he says about that too!