How many sounds I used actually to teach depended on the learners’ native language. With Czech speakers, for example, I would spend nearly as much time on aspirated /p/ as on most of the other phonemes together.
Much of my work on pronunciation was devoted to word and sentence stress, chunking and intonation, plus the usually small number of phonemes that the learners had difficulty with.
If you are asking about how many sounds I wanted my learners to produce as accurately as possible, the answer is the 43 phonemes here, plus non-phonemic /i/ and /u/, syllabic /l/, /m/ and /n,/, and the glottal stop. I did not worry particularly about /ʊə/, as many native speakers don’t use it. If necessary, I would work on allophones such as aspirated and unaspirated /p/ and clear and dark /l/.