The phonetician J. C. Wells in his Accents of English (1982) gives an historical account of the phenomenon known as yod-dropping, which can be heard in the pronunciation of new as /nu/ as opposed to /nju/. I'll try to sum up his views by providing some excerpts from his book:
Pairs such as threw-through, brewed-brood used to be distinguished by the use of a diphthong in the former member of each pair as against a monophthong in the latter. The diphthong of the /ɪŭ/ type, developed into a rising /ju/ through the transfer of syllabicity from the first segment to the second; in certain environments the /j/ then disappeared, a development we may refer to as early yod dropping. The pairs mentioned are accordingly homophones in most accents.
In London the falling diphthong /ɪŭ/ had by the end of the 17th century given way to a rising /ĭu/, phonetically identical with the inherited /ju/ of youth etc. This development brought into existence a large number of new initial consonant clusters involving /j/. Several of them were intrinsically awkward to pronounce, so that we find that from the beginning of the 18th century the /ĭ/or /j/ element disappeared in certain environments, leaving a vowel identical with /u/ and creating new homophones such as threw-through.
The environments in which early yod dropping applied most generally are (i) after palatals (including palato-alevolars), as in chute, chew, juice, yew; (ii) after /r/, as in rude, crew, shrew, grew; and (iii) after consonant plus /l/, as in blue, flue, blew, glue.
The accents of East Anglia are notable for having extended yod dropping to most or all postconsonantal environments, for example in few /fu/, music /muzɪk/, cube /kub/, Hugh /hu/.
Other accents occupy intermediate positions, retaining /j/ after labials, velars and /h/, but perhaps not after some alveolars.
In RP there is variability in the environment of a preceding /θ, s, z, l/ as in enthusiasm /ɪn'θ(j)uzɪæzm/, suit /s(j)ut/, lewd /l(j)ud/; the yod is consistently retained after /n/, as in /nju/ new, and also after /t, d/, as in tune /tjun/, duke /djuk/ (where there is the further possibility in casual speech of yod coalescence to /tʃun/, /dʒuk/).
In General American, and also in part of the south and midlands of England, /j/ is lost after alveolars /t, d, n, l, s, z/ but not after labials or velars. In these environments General American predominantly has plain /u/, thus tune /tun/, duke /duk/, new /nu/ etc. Some easterners and southerners, however, have either /ju/ or the diphthong /ɪu/, and General American is not entirely uniform.
For an example of yod-dropping after labial, labio-dental and velar consonants, I re-link the video mentioned in one of the comments above where an East Anglian accent can be heard :