(By the way, had you run into the geeky AmE "snarky" to mean sarcastic? I'd always wondered where that word had come from, and now I think I see a family resemblance.)As I said there, I love the word snarky because I find it rather evocative. But there are a couple of assumptions to challenge in Jo's query. First, it doesn't seem to be exclusively AmE--the first OED example of it is from the very English book The Railway Children. It comes from the dialectal verb snark, meaning 'to snort' and also 'to nag, find fault' (which has some cognates in other Germanic languages). AmE speakers may use it more commonly than BrE speakers these days, or it may still be regional--I don't know--but these may be reasons why Jo assumes it's AmE.
Second, it doesn't quite mean 'sarcastic', like BrE sarky, though it could readily be used of someone who was being sarcastic. It means something more like 'irritable, bad-tempered' (OED). If someone's being sarcastic, it's often a symptom of bad temper, so one can see how the two have come to be linked in (some of) our minds. An AmE word that comes to mind is snit, which means a little fit of bad temper. I wonder if the case could be made for some sound symbolism between /sn/ and bad temper. /sn/ is onomatopoetic in words for nose-breathing-actions: sniff, snort, etc. And bad temper is getting one's nose out of joint or possibly turning one's nose up at something (and we get /sn/ in snob...). [There seems to be at least one academic paper on the topic, so I won't go any further on this...probably not news.]
Now, a BrE speaker may be led to believe that snarky is AmE because they're more accustomed to (BrE) narky, which the OED gives as a synonym of snarky. This is derived from to nark 'annoy', hence (BrE) narked 'annoyed'.
Now, I was surprised to learn that the 'police informer' sense of nark is related to this. It comes from a sense of nark meaning 'nose'--so a nark noses around for the police. But in AmE we also have narc, short for 'narcotics officer'. I always believed that the informer sense was based on narcotics too. This is why one shouldn't make assumptions about etymologies based on the apparent similarities between contemporary words. It's likely that narc was influenced by nark, and that narky, snarky and sarky have influenced each other. Still, they have different roots.