[denethorthefirst]

Tolkien wrote that the Orcs of Mordor spoke a "debased" Version of the black speech (the real-world equivalent would probably be the difference between classical latin and vulgar latin). For all we know even during the time of the war of the ring the black speech was still the official language of Mordor and in extensive use throughout the country (at least among the bureaucracy and the administration of the country).

As to the Dragons: they are portrayed as highly intelligent and long-lived so i would guess that they are able to learn and master several languages. The mother tongue of them is probably the one that was spoken among the higher ups in Angband when Morgoth created the Dragons (some twisted version of Valarin maybe?).