Continue, persevere are formal (from Old French) correct?
They are very common in conversations in French (maintenant je continue mes lectures, tu dois persevere pour arrive)
Are they never said in speaking conversations in English at all?
When I say “continue” (familiar to me because I use them every day in French) in a English conversation the English speaking person seams a little bit surprised.
In which cases “keep on” or" go on" are correct and I wonder in which cases “press on” is more accurate.
Press on is typical to the British Isles and Keep on is used much more often there too - both indicate moving forward through a difficulty, keep on became very popular during WWII and has had a recent rise in popularity. Go on is often used on both sides of the Atlantic but a bit differently, for North America it is often used to encourage the telling of a story or event, in Europe it is more of a “off you go then” which can apply the telling of a story to to a task.
Persevere has more of a sense of getting through something difficult and long term. Continue had more of a sense of time passing - such as resuming after an interruption or also to go the finish, “Now I will continue with my reading” would be correct. 'Continuing Studies" in North America indicates courses one takes after graduating from University for further development.
I wouldn’t say that either are formal. Continue is a very common word. Persevere is a less frequently used word but it can still certainly be used in conversational English.
“If we want to continue improving our language skills, we must persevere!”