How correct and how widespread is the usage of writing pinyin "ü" as "yu"?
It's a workaround, and I would recommend that it be treated as such.

The official romanisation of Chinese has an international standard, currently (as of mid-2019) ISO 7098:2015. This does mandate the use of pinyin ü where required.

However, the article refers to a PRC national standard, GB/T 28039-2011, for "acceptable" (not correct) romanisation in international fields (it gives international sports tournaments as an example). Toneless pinyin, how to do double-barrelled surnames, and how to deal with a lack of ü are all covered. For the latter, it specifically states under section 6:


  根据技术处理的特殊需要，必要的场合 (如公民护照、对外文件和书刊等)


This was extended by the website above, but with a rule to transliterate 略 lüè and 虐 nüè not to lyue and nyue respectively, but to as lue and nue (as doing so does not introduce any ambiguity anyway).

Why ü as yu? That's the most sensible option for the sound represented in IPA by /y/, if you exclude all other diacritics:


The "ü" sound in Hanyu Pinyin is the same sound as "yu". 女, with "ü", rhymes with 语, with "yu".
Tongyong Pinyin, official in Taiwan between 2002 and 2008, uses "yu" consistently instead of "ü".
The Jyutping romanisation for Cantonese (which also has the /y/ vowel frequently) uses "yu" consistently too.


In what circumstances would one see this then? Passports and other "inherently international" official ones seem to the main target, where known technical issues exist. I personally would consider this workaround to be less correct than the use of ü (as per international standards), but if you are writing an article with a specific company, they may be using that online source as a sort of "house style".

In the wild, of course, the most common is in practice "v".