KWIKS-'t is also dying out. Almost everyone except me now says ke-HOAT-ay or ke-OAT-ay. (Unless they go for holidays on the Costa Brava and imagine they can trot out a KH.)

I say it the way it's spelt: KWIKS-'t.

In Spanish in Cervantes' day it was ki-SHO-te and so spelt, Quixote. The letter X represented SH as it still does in Basque, Portuguese, and Mexican languages.

This pronunciation was taken into French, and given a French spelling Quichotte. (Since then the final -e has become silent in French.)

The spelling was taken into English, but was pronounced in an English way, as in 'quixotic'. Attempts to render a Spanish pronunciation are only modern.

In Spanish the SH sound changed to KH. The normal source of this sound was Latin GE, GI. The path was something like G > J > ZH > SH > KH. This sound was spelt GE, GI, JA, JO, JU.

But there were also SH's that didn't come from Latin -- e.g. Xabier, a Basque name -- so those were spelt with X. In Portuguese X from Latin X can have this pronunciation, so I imagine that was also a source of the Spanish SH sound, but I'd have to look that up.

However, once both sources gave the same SH sound and this changed to KH (the modern pronunciation in Juan, jota), the words were respelt with G or J instead of X. So the modern Spanish for the name is Quijote. (So also Javier < Xabier.)

Therefore Quixote is an English word and we can pronounce it as it suits us.

Since you asked.