In what language did Caractacus deliver his speech to the emperor Claudius?

Caradog's father and uncle were educated in Rome at the Court of Augustus - Problem solved.

The tribunal of Caractacus was, according to Tacitus, a huge and celebrated affair, with an assortment of prisoners from the 9-year campaign against the Britons. To transport all these men and women from Britain and arrange for the parades, etc, would have taken months at least. This long captivity would have given Caractacus the opportunity to learn Latin. Also, the speech is short and not very complex, so he could easily have learned what he needed to know to speak it.

Here is the speech as given by Tacitus:

Si quanta nobilitas et fortuna mihi fuit, tanta rerum prosperarum moderatio fuisset, amicus potius in hanc urbem quam captus venissem, neque dedignatus esses claris maioribus ortum, plurimis gentibus imperitantem foedere [in] pacem accipere. praesens sors mea ut mihi informis, sic tibi magnifica est. habui equos viros, arma opes: quid mirum si haec invitus amisi? nam si vos omnibus imperitare vultis, sequitur ut omnes servitutem accipiant? si statim deditus traderet, neque mea fortuna neque tua gloria inclaruisset; et supplicium mei oblivio sequeretur: at si incolumem servaveris, aeternum exemplar clementiae ero.

Translation: Had my moderation in prosperity been equal to my noble birth and fortune, I should have entered this city as your friend rather than as your captive; and you would not have disdained to receive, under a treaty of peace, a king descended from illustrious ancestors and ruling many nations. My present lot is as glorious to you as it is degrading to myself. I had men and horses, arms and wealth. What wonder if I parted with them reluctantly? If you Romans choose to lord it over the world, does it follow that the world is to accept slavery? Were I to have been at once delivered up as a prisoner, neither my fall nor your triumph would have become famous. My punishment would be followed by oblivion, whereas, if you save my life, I shall be an everlasting memorial of your clemency.

Since Tacitus probably made it up, like most speeches put in the mouths of historical figures by ancient historians, the answer would be "Latin".

Addressing your last statement, by 50ish AD Roman and the Celtic languages were not mutually intelligible.

It is true that they are related sets of languages. Exactly how closely is up for debate, but the debate among linguists is essentially over whether they split into their own languages around 3000BC, or instead around 2500 BC. The 3000BC number is the commonly-accepted one these days, but even if you take the more recent one as Italio-Cetlicists do, there's two and a half millennia of evolution between the languages spoken by Caractacus and Claudius. To give perspective, the ancestors of English and German were mutually-intelligible 1300 years ago.

As for the ability of a tribal leader to give a moving speech in a civilized conqueror's own language, this should not be surprising to anyone familiar with American History, as it happened regularly with Native Americans and the US government. This feat is probably helped by the fact that tribal leaders are generally chosen for their oratorical skills, while leaders in more highly organized societies typically derive their legitimacy from wealth or parentage.

In other words, its quite likely Caractacus was the most talented orator his society had to offer.

While it might have been temporarily effective here and there, in the long run these verbal appeals helped the Celts about as well as they helped the Native Americans. Poor farmers hungry for land have a force not even their nominal rulers can stop.

Being that Britain had been exposed to Roman influence for close to a century - Caesar having made first contact with his invasion around 55bc. There was constant diplomatic and trade relations between the British and Romans following that.

As Caractacus was a member of the ruling class, it's entirely possible that he spoke Latin to some extent.

As mgkrebbs points out, a translator is the other likely option!