The problem here is that Y is NOT a cononant in cases like the one you (and many answers) give.
There IS some confusion about this letter. That is simply because, unlike a,e,i,o and u are ALWAYS used to mark vowel sounds, but Y marks a vowel sound in some situations, a consonant sound in others, so the list of vowels has to add "SOMETIMES Y". But this shouldn't be too difficult, because it's pretty clear in any given word which it is. Y is a consonant ONLY when it appears at the BEGINNING of a syallable (esp at the beginning of words -- 'yes', 'yearn'...). At all other times (middle or end of syllables), which is much MORE often, Y stands for a vowel sound. It Modern English that vowel sound is usually exactly the same as "i", thus the Y in "sky" marks the same vowel sound as the "I" n "skies".
This means that your example, "cyst", is pronounced EXACTLY the same as it would be if written "cist". So obviously the Y is a vowel in this case, and in all the others people have listed.
Some try to say that Scottish words like "cwm" have no vowels. But that is making the same mistake. W works much like Y. It is used as a consonant at the beginning of syllables, but otherwise it is a vowel, the equivalent of the related letter "U". Now in ordinary English words the vowel use of w only happens in combination with other vowels (aw, ew, ow), but it should be clear that it IS a vowel sound since the same combinations with U are pronounced the same way. See the following pairs: paw/pause, how/house, few/feud. (Note that W is most often used at the END of root words; in the middle we are more likely to see the U.) In Scottish W can appear as a vowel all by itself, again with the value of "U" -- thus "cwm" is pronounced like "coom".
So Y and W won't help you much. As a matter of fact the ONLY English words that lack vowels are interjections, esp. words that imitate sounds we make. The main ones you will find (several of them in dictionaries) are the following:
If you want more, "The Dictionary of Consonant-Only Words" by Craig Conly is an entertaining list of many consonant-only words (mostly made up) which have actually appeared in literature:
(He lists 105 words under "T" alone! )
If you need more on the distinction between consonant and vowel see: