Potatoes never grew and rotted

If the soil has dried out enough, yes, you could still plant a late variety. I'm not familiar with UK varieties but a retail shop would know what you could plant. I wouldn't even bother to chit them, just plant asap.

As Bob Flowerdew says, if you plant potatoes late, then plant an early variety.  Early varieties have a shorter time to maturity.   

I haven't seen any seed potatoes on sale anywhere any more.   So find a bag of Reduced to Clear potatoes from your supermarket, if you can, preferably one that is an organic variety.  The organic varieties are more resistant to diseases  (scab, blight, blackleg, you name it, potatoes encounter every disease going)., even though they might not have the cooking qualities of the King Edwards and the Maris Pipers, blah, blah.   Also, they will not have sprayed the potatoes with any sprouting inhibitor chemicals, so they will sprout more quickly.   

Yes, you can introduce lots of potato diseases by planting supermarket potatoes instead of seed potatoes, but if the ground you are using is an allotment, or near one, I doubt you will introduce any disease that isn't already there.