There are lots of possible implementations of phonetic searching. Limiting based on query term count would save lots of overhead, and limiting it to terms that aren't in the index (or have very very low counts) could work, too. These are things we could test beforehand, to see what the expense and benefit of computing various things work out to be.
Soundex is pretty old, but it works okay. It's easily modified to be a bit smarter, too. The baseline implementation only considers the first few consonants to maximize recall for genealogists who are willing to sort through lots of hay to find that needle. Double Metaphone seems to be out there and available (may require a consultation with a lawyer), while Metaphone 3 is clearly for sale (the license is pretty nice as long as you don't want to share it).
As for using it with other languages, hmmm, I have to think. The phonetic "index" is generally would not be directly searchable in normal text; it isn't a phonetic representation of the word, it's just a code that similar sounding words tend to have.
Phonetic spelling comes in a few varieties on enwiki. There are IPA spellings[1] and dictionary style phonetic spellings. The dictionary spellings can have different conventions (I don't know how well standardized they are on enwiki—linguists have been pushing for IPA since it is standardized). But even IPA can have differences of detail that make it unsearchable. Gorbachev has three IPA pronunciations: /ˈɡɔrbəˌtʃɔːf, -ˌtʃɒf/ in English, and ɡərbɐˈtɕɵf in Russian. The first one includes primary and secondary stress information, the second one is only the last syllable of the name, and the third one has primary stress info. Leaving any of the stress info out, or try to search for the second pronunciation, and you don't get a match. So, I don't think we can leverage the phonetic spellings that are in articles.
However, it would definitely work for reasonable spellings of many words of non-English origin. Possibly aparrachick for apparatchik, probably shadenfroid for schadenfreude, but probably not paree for Paris (there's already a redirect for that, though!). It depends a lot on the spelling system of the source language (French has too many silent letters, for example) or the transliteration system used, and the history of the borrowing (when spelling and sound don't match up, English tends to keep one and adapt the other, which is good, but sometimes it turns weird).