Hmmm… I think British English wins here because we say ‘crush’ too but it means far more than fancying. It’s kind of an old-fashioned word here (I never hear anyone under 40 say it) but ‘crush’ denotes a chronic infatuation rather than just thinking someone is fit. Fancy can be exactly like the old-timey thing of ‘a passing fancy’ whereas ‘crush’ is more serious.
In posh female circles decades ago, the word ‘pash’ (as in abbreviation of passion) was used instead of ‘crush’ and it nearly always meant a lesbian fancying, it was a cute term. Don’t hear it much now. It’s the kind of thing Diana Rigg would say. 😛
This reminds me of the current stages of sexual pair-bonding. In the East Midlands of England where I live, this currently goes:
* shagging / fucking / sleeping with someone
* seeing someone
* going out with someone
The first two phases are assumed to be non-exclusive whereas the last is commonly assumed to be the establishment of heteronormative monogamy (even in non-het relationships).