The symbols used to represent vowel phonemes are only rough approximations of the actual quality of the of the vowels themselves. We need to have some sort of symbols to represent the vowels in a language when doing phonemic transcription. The symbols in phonemic transcriptions do not represent the specific idealised sounds represented by the symbols in International Phonetic Alphabet. We usually simply take a symbol that represents a sound quite similar to the ones we find in the language we are transcribing in. We usually also take into account other factors, for example the writing of the language that we are transcribing. The standard convention for representing this vowel in English Phonemic transcriptions is /aɪ/. Most phoneticians and phonologists, language textbooks and dictionaries of British English use this symbol.
Now the actual vowels produced by speakers of British English change slowly but constantly over time and differently in different regions. Because of this, some of the symbols used to describe vowels may now be describing a sound that is quite different from the quality of sound it describes when used in the International Phonetic Alphabet.
For this reason, very unfortunately, the Oxford English Dictionary hired a man called Clive Upton to adjust the symbols used in the Oxford Dictionaries. However, I think it is fair to say that the general feeling amongst phoneticians and phonologists working with Southern Standard British English is that most of these adjustments were not very good at all, and some were actually very bad. The change of symbol from /aɪ/ to /ʌɪ/ is one of the changes that is heavily criticised for being far more misleading than the original symbol. What we need to realise is that the symbol is not a close representation of the vowel, and that whichever symbol we use, it doesn't affect the sounds that people make when they use this phoneme. Nevertheless, /ʌɪ/ is a very bad approximation of this vowel sound. The starting point for this vowel really is nowhere near the [ʌ] sound represented by the symbol in the International Phonetic alphabet. Because of this almost everyone has ignored the changes Upton made to the transcription system used in the OED. Professional phoneticians, dictionaries and published papers still use the original /aɪ/ symbol.
If you're interested in the system we use for transcribing RP English, then you could have a look at this excellent page written by eminent phonetician Professor John Wells, author of the Longman's Pronunciation Dictionary. It has some very interesting comments on Upton's scheme used in the OED.
