I searched the internet and Dio's source, but I can't find out how the Romans transported the elephants to Britannia. They must've used boats, but how do they load elephants onto boats, and what kinds of boats could support that weight? And are we supposed to believe that these elephants were brought from Africa all the way up through Europe to Britannia? Is there any other evidence that elephants were brought to Britannia?
The English Channel, at its shortest point, is about 21 miles. Hannibal brought his elephants over the Strait of Gibraltar, which is under 8 miles at its shortest point.
Elephants had been transported over comparable distances before. Pyrrhus transported twenty of them to Italy in 280, and the successors transported elephants by sea all over the Aegean. Pliny mentions African elephants in the beast-hunts during the games--presumably these were transported by sea. Diodorus Siculus mentions deep draft ships purpose-built for transporting elephants, which are mentioned in a few other places. We don't know what these elephant-transports were like, but they're hardly surprising--horse-transports are attested pretty early on in Greece, and elephant-transports are a pretty logical step from there
EDIT: So I got so caught up in finding references to the transport of elephants by sea that I forgot an important point, which is that pretty much nobody thinks Dio's right. Luckily /u/Tiako mentions it in his comment. There's no good reason to think that Claudius actually did this, especially since no other source agrees with Dio--generally if something that Dio says sounds silly and isn't in an earlier source it's probably not really true. Logistically it's possible, such things had been done before, but probably not
It isn't mentioned in any of the earlier sources (Tacitus and Suetonius primarily) and it is generally not considered likely. Although the English Channel is not particularly long, it is somewhat temperamental in terms of weather and currents and the English coast rather famously does not have very many good landing points. Claudius certainly had more information at his disposal than Caesar, but you can read Caesar's accounts (in DBG book VI) of his landings to get a sense of how much of a pain the crossing can be. None of this out and out disproves the use of elephants, but the source for it is late and unreliable, elephants weren't really a standard part of the Roman army at this point, and they would have been a logistical headache.
Would these elephants have been of the existing African species (Loxodonta africana) ? Or would they have been of the now extinct North-African subspecies (Loxodonta africana pharaoensis)?
The latter were smaller with a natural habitat further North, so that alters the logistics somewhat.