Pronunciation of "sue"
In many accents of English, /sj/ at the start of a syllable has been simplified to /s/. This simplification has progressed further in North American English than in British English, but it's gone pretty far in both. The pronunciations aren't neatly divided between different dialects of English: there may be some Americans that use /sj/ in sue (I know that some Americans use /nj/ in new), and there are certainly Britons that use /s/ in sue. One of them is not "correct".
/sju:/ is the pronunciation of sue in accents without this simplification.
/su:/ is the pronunciation of sue in accents with this simplification.
This is not a matter of words with a stressed /u:/ sound "acquiring" /j/ in some accents. Words such as soon and soup are pronounced with /su:/ rather than /sju:/, even in accents that use /sju:/ in sue, suit, or super. It's just like the contrast between /buː/ in boot and /bjuː/ in beauty that exists in all widespread accents of English (although not in some regional accents of England, where /bj/ at the start of a syllable has been simplified to /b/). The simplification of consonant clusters with /j/ is called "yod-dropping", after the Hebrew letter yod/yodh which represents /j/. Words that historically contained /juː/ are typically spelled with u, ew or eu.
The name Susan traditionally started with /sju:/ not /su:/, so there is no underlying reason why sue and Sue should be anything but perfect homophones. However, some speakers show variable simplification of /sj/ to /s/, where only some of the words that historically had /sj/ are pronounced with it. It is conceivable that Sue and sue might be distinct in this way for some speakers, but I doubt it is systematic.
The entry for U in Fowler's Dictionary of Modern English Usage from 1926 noted that /sj/ already showed a tendency to be replaced with /s/ in southern British English of the time:
After s & z there is a tendency to convert the orthodox ū to o͞o or o͝o, e.g. in superior, Susan, supreme, suzerain, suicide, suet, suit, presume, Zulu ; this class is comparable to the lu words, but the decline of ū is far less marked.
Fowler uses ū, o͞o, o͝o to denote /ju/, /u/, /ʊ/.
As Robusto mentioned, some words have a stressed "u" sound. Sue in British-English has two main definitions; 1/to institute a process in law against; bring a civil action against:to sue someone for damages.
2/ a noun, a female given name, Susan, Susanna.
From 40 years+ of my lived experience in England, definition 1 is pronunced with a stressed "u", similar in sound to few.
Definition 2 is pronunced with a softer "u" sound similar to moo or loo. The two words sound so similar that only context could offer a discernable distinction between the two words amongst the most native of English speakers.