First, I seem to recall that he arrived in the US around 1922 in 1920. While it's possible for an immigrant to the US at that time to have lived and worked fully within their respective immigrant community, the fact that he found success as an actor outside the Hungarian community during those nine 11 years suggests to me that he did learn English with reasonable proficiency.
Second, he had already played Dracula in the stage play on which the 1931 film was based, while the story I'm questioning makes it sound as though the film was his first attempt at the role.
Third, there was a 1932 series of interviews with Hollywood stars that included him, and he speaks English without apparent trouble in it. At the same time, I recall it seeming somewhat scripted, since it seemed overly calculated to reinforce his sinister image. Are there any more clearly unscripted interviews with him from that period?
EDIT: Found the 1932 interview. It doesn't seem quite as scripted as I thought, at least not in its entirety. He uses "sculpture" as a verb, and while it is not incorrect to do so, I think most native English speakers would instead say "sculpt" in that context; this suggests to me that the things he says in the interview are largely his own words.
Lugosi got in trouble for being making socialist comments in the very brief period after WW1 where it was acceptable (the Communist politician Béla Kun held power from 1919 to 1919 before being deposed by the conservative Miklós Horthy), so he left Hungary for Vienna and then to Berlin. He tried working in Germany briefly but there was already an overabundance of actors seeking work, then worked his way over to the US on an Italian cargo ship, winding his way up from New Orleans up to New York.
He started in some plays for Hungarian-speaking groups, but both the audience and attendance was quite small. He managed to get his break in the US (as recalled in a later interview) by a theatrical manager (Henry Baron) asking him to play in The Red Poppy. Lugosi admitted he did not know English, but convinced Baron to get him a tutor (with salary paid out of his future earnings).
For The Red Poppy (and possibly some of his other early work) he did apparently go the phonetic route, learning the words like they were to a song, but his English did start to improve after. However, in the years leading up to the play version of Dracula he was still living in the Hungarian community, so he hadn't quite got mastery; in 1928 he gave an interview that said he had his dialogue first translated into Hungarian to be able to think "it out as though I were going to act it in my native tongue" and that "when I begin to feel at home in it, I begin to concentrate on the English diction, so that I may speak the part as it should be spoken for American audiences."
This quote is what gave the impression he was still learning things phonetically, and the myth that he "had no English" at the time. However, he gave that interview in English! Lugosi also had no trouble with English in a 1927 conversation later recalled by horror director William Castle (1927 also being the year the play version of Dracula opened), and shooting in 1931 for the movie Murders in the Rue Morgue the director Robert Florey said that Lugosi
didn't have to learn his lines phonetically because when ever our producer sent us some 'yellow pages' on the set at the last minute Bela would concentrate on his new lines for a few minutes, ask for a last rehearsal and then was ready for a shoot.
So it is definite for the film version of Dracula that Lugosi did not have to learn things phonetically, although he was still conscious of his accent and would ask the sound man to warn him if his accent had ever got too thick on a particular scene.
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Kelly, K. E. (1996). Lugosi in Hollywood: A Hungarian Actor's Rise and Fall as a Movie Star. Hungarian Studies: Hs., 11, 115.
Lennig, A. (2013). The Immortal Count: The Life and Films of Bela Lugosi. United States: University Press of Kentucky.