Since you found these two words, I guess you know what they mean. This could be helpful if you ever ran across them. Of course, with a good working knowledge of English, and given the context, you could probably guess the meaning. In fact, I don't recall ever encountering them before your question. So on the level of frequency, they are extremely low. Some thesauruses include them, but this is just to be complete.
To answer your question, the likelihood I would *ever* use them is <1%.
The Oxford English Dictionary (OED) gives this sentence from 2001:
"Scorsese was increasingly adulated by critics and swept up in the Hollywood money machine." (R. Carney: "Cassavetes on Cassavetes", xi. 469).
However, besides that, the OED provides only one sample sentence in the 20th century for 'adulate' and that is from 1906. 'Belaud' is even less used, and the OED gives only 4 examples, the latest dated 1882.