I think there are many approaches. For example, my understanding is that professional movie actors often use dialect coaches. For those not willing to go to such an expense, you might consider not just studying the phonemes but their phonetic realization, particular with respect to vowels.
You can see an example of varying phonetic realization in this part of Wikipedia for the British "Received Pronunciation," "General American," and "General Australian." Please realize, however, that there is tremendous variation, particular among UK and American accents. See here for an example of the complexity. There are also differences in intonation patterns not captured at the level of consonants and vowels.
As a native speaker of "General American" English, more or less, I immediately notice the difference between the two audio samples you linked to. I can also reproduce either version at will. Even so, I would transcribe them with the same phonemes. On the first cite I linked to, however, the /ʌ/ is described as "short" in the Received Pronunciation and "lax" in General American. This difference is also what I hear. The different realizations of /ʌ/ also fit into a different vowel structure within the dialects. If you study these differences and can reproduce the difference between "short" and "lax," you can begin to adjust your pronunciation and your ear.