Are there any English words that have no vowels and no Y?

Brrr: The sound of shivering
Crwth: An ancient stringed musical instrument
Cwm: A cirque (a steep-walled mountain basin shaped like half a bowl)
Grr: The sound of a dog
Hm: An interjection expressing assent
Hsh: An interjection used to urge silence
Nth: adjective pertaining to an indefinitely large number
Phpht (pht): An interjection used to express annoyance
Psst (pst): An interjection used to attract someone’s attention
Shh (sh): An interjection used to urge silence
Tch: An interjection expressing vexation or disgust
Tsk: An exclamation of annoyance
Tsktsk: To utter tsk
Tst: An interjection used to urge silence

That's all I could find.

"Rhythm" and "syzygy" are the longest English words without vowels. But they do have Y functioning as a vowel.

In the UK, I don't think "y" is not considered as a vowel as it is in North America. This fact has caused many arguments at our Pub Quiz nights

My crossword favourites are "tryst" or the ubiquitous
"gry" but there must be many more.

If we refer to words that lack "true" vowels (a, e, i, o, and u), the answer is that a number of them do; among them are: hymn, rhythm, myth, sylph, and syzygy (the alignment of three celestial objects). These words contain the so-called semivowel y and none of the true vowels.

http://www.grammarmudge.cityslide.com/articles/article/1029289/8966.htm

One might dispute crwth and cwm, as they are welsh words, and w is a vowel in welsh.

Rhythm and syzygy do contain vowels. Y in English can be both consonant and vowel. If it's pronounced as an i then it is a vowel. Some grammars follow the Latin rules for English, and the Latin rules don't work for English. And they forget that in Latin v could be consonant or vowel (we usually write the vowel sound of v as u) and don't allow English the same privilege with regard to y. As Dave points out, the w in crwth is a vowel in Welsh, and therefore must be when the word comes over into English as a loanword.

The answer depends on exactly how you define an "English word". Since Scrabble claims to allow only English words, and my downloaded game allows such words as mm, I would have to say yes. (But they are not 'normal' words, as discussed in the following extract from [grammarmudge.cityslide.com/articles/article/1029289/8966.htm, no longer online])

The debate is further complicated by the view held by some people that certain uttered sounds that have distinct meaning (e.g., Shhh! Grrr! Psst! Zzzz!) are true words and therefore qualify as words without vowels (or, for that matter, semivowels). If we disqualify these words and words ending in y (together with acronyms such as RSVP and abbreviations such as P.S. and P.M. ), every English word contains at least one of the principal vowels.

After I posted the above answer, several people disagreed. Among the dissenters were those citing cwm, a Welsh word for "valley," which does have a listing in some English dictionaries. Therefore, I probably needed to clarify further to rule out words that are strictly dialect, especially Gaelic or Welsh words in which w may have a vowel sound. Someone else suggested that nth was a word with no vowels. But the n in nth stands for any number, so then we might as well argue that 1st, 2nd, 3rd, 4th, and so on are vowel-less words.

A crossword staple: nth. As in, the generic version of eighth, ninth, tenth, etc.

http://www.thefreedictionary.com/nth

here you go.
http://wordslisted.com/words/no-vowels/

Scrabble Words – No Vowels, No Y
BRR
BRRR
CRWTH
CRWTHS
CWM
CWMS
HM
HMM
MM
NTH
PFFT
PHPHT
PHT
PSST
SH
SHH
TSK
TSKS
TSKTSK
TSKTSKS

I think it depends on your definition of 'vowel'. Cwm and Cwth which are borrowed from Welsh have the 'w' producing a vowel sound or (beautiful word) monophthong.

https://en.wikipedia.org/wiki/Vowel

It should be noted that the reason y is sometimes included as a vowel is because it often stands for a vowel sound. Thus, if you want to exclude y, you may want to exclude w for the same reason: it represents a vowel sound in a few arcane words like cwm amd crwth.

Excluding both y and w, most words that do not contain vowels are mainly interjections like bzzt, grrr, hm, hmm, mm, mmm, pfft, pht, phpht, and tsktsks.

Wyrm and myrrh can be considered vowel-free as the y in both of them does not stand for a vowel sound. On the other hand, nth is pronounced with a vowel sound ("enth") in it.

https://en.wikipedia.org/wiki/English_words_without_vowels#Without_W_or_Y