I'm afraid I don't know what the odds against someone having an elderly relative who owns a little white dog are, but even if I did I don't think it would change much. You would also have to demonstrate that:
-- there were no physical signs that might have led to the woman who said it (e.g. if nungate had recently visited her father there may be a stray dog hair or two);
-- the woman did not already know the information, or was not made aware of it some other way just prior to saying it;
-- nungate was the only person she gave that reading to, or at least not the only person she gave that reading to correctly.
And probably many other things besides. Even if all of this turned out to be true, other questions that would need answering might include: how did the woman say this -- all at once: ", ah yes, your father has a little white dog", or somewhat slower, "I'm seeing a dog... small... white ... your father's?", at each step waiting for some subtle visual sign of confirmation that a lot of us can't help giving away.
It doesn't necessarily apply even 15 years ago now, but also a lot more people know each other without having met or even discussed it than ever before. I was freaked out for a while when someone I'd never met before asked me how the Listener crossword that day was going. ?!?!, I thought. Turns out that he'd seen me on UC, seen me say that I was on UC on this site, and matched that name to my posts in the Crosswords section on AB. Nothing mysterious going on there -- I've revealed a lot about myself which any competent stalker could find if they wanted. Granted, this is something that you couldn't really do before the days of the internet as easily -- but that is yet another thing to be ruled out.
I think Derren Brown for example is able to do a lot of this sort of thing, but acknowledges that it is of course a trick, or at least nothing mysterious or as yet unknown. I'm inclined to believe that all clairvoyants/ mediums/ fortune-tellers are relying on the same techniques he uses, or at least similar ones, until someone proves otherwise.