10 half lives of U-238 will pass in around 45 billion years. We have only about 5 billion years before the sun starts expanding beyond Earth's orbit. Therefore, the answer to your question is "inside the sun".

Now, the main part of your question:

Of course this is a ridiculous argument because U-238's half-life is measured in billions of years

You are entirely correct. We can even calculate that. Let's ask Wolfram Alpha how much uranium will have decayed after 1000 years. This is the answer:

remaining fraction of number of particles | 99.9999845% = 0.999999845

So for all practical purposes, after one thousand years (!!), the amount of uranium in the Earth does not change (other than the uranium used for power generation).

I couldn't tell her whether or not the world's reserves were shrinking in a geologic time scale

First, note that the word "reserves" has a very specific meaning. You are probably asking simply about the quantity of uranium in the Earth. So yes, it is shrinking. The half-life of uranium is about 4.5 billion years, which happens to be the age of the earth. So the Earth currently has half of the uranium it had when it first formed.

I believe there may be a handful of isotopes that decay into it

Yes, Pu-242 will decay to U-238 with a half-life of 375000 years. Since we're more than 100 half-lives past the formation of the Earth, there is no Pu-242 left on Earth for all practical purposes. Statistically speaking, there might be a few atoms left hanging around since then, but don't put your bets on their decay for increasing the amount of U-238.