In short, no, Weltschmerz is not untranslatable, but it’s often rather hard to translate well into English.
The real question is either whether there are any words or phrases in any language that are untranslatable or whether there are words in a particular language (e.g. German Weltschmerz) that have no direct translation or cognate in a certain other language (i.e. English).
The meaning of a word (or a linguistic sign) does not come from a dictionary, but – as famously coined by Ferdinand de Saussure – always from the side. That means it matters a lot who says what where, how and why to whom, because verbal communication is more than linguistic transmission of information. All parts together constitute the actual and individual meaning.
That’s the reason for why so many people claim that one should preferably read a book or watch a movie in its original language to really get the most out of it, also why there is literature study and why students have to research an author’s biography and culture of the target audience to properly and deeply analyze any literary work. Anyhow, a translation can be a creative act all on its own if it goes beyond mere word and phrase mapping. This is even harder if other metrics have to be met, e.g. the rhyme scheme in a poem or synchronous lip movement in a dubbed film, or if there’s a pun to be kept.
As has been mentioned in the comments, Weltschmerz may have a close equivalent in Portuguese, saudade, but none in English (yet or anymore). Its general essence can still be provided in few words, though, and that’s what German-English dictionaries do. Often they will list several meanings, e.g. “1. world-weariness … 2. causeless sadness …” and one should know and consider all of them for optimal understanding. These descriptive meanings in another language may be proper choices in some contexts for translations into said language, but more often than not the best translation requires a more complex approach and may result in completely different words and grammatical constructs.
Take Gurke and Affe for instance. If one needs to, German is able to differentiate between cucumbers and gherkins or monkeys and apes, but it does so with compound or adjective attributes instead of separate lexemic stems as in English. Linguists (and philosophers and psychologists) differ in their opinions (and findings) of what such distinctions in language mean for paths and limits of thought.
Anyhow, if an abstract concept has an established label in one language but not in another, this may be adopted as a loan word, weltschmerz, or inform a neologism, e.g. by literal translation, *world-hurt. These will carry and further develop their own meanings, however, which are not identical to all the possible ones in the source language, cf. angst, which will also change over time (e.g. Welt- could become a generic elative prefix in German like Super- / Riesen-, so there would then be an alternative meaning of ‘major pain’).