In the States, we use the term "pharmacy" or "drugstore," but what is the equivalent in the UK? I checked two sources, but came up with nothing.
It's a chemist's, though drugstore is starting to creep into standard UK usage too.
Chemist, chemist's, chemist's shop or, sometimes, pharmacy. I've never heard "drugstore" in the UK, though one of the big chains is called Superdrug.
Technically a chemist's [shop] will contain a pharmacy, which is the counter where prescription drugs can be obtained. Because it may be necessary to get these when shops are closed, there is a rota (published in the local paper, for example) of out-of-hours pharmacies; some of these are in supermarkets or 24-hour shops.
I think that the answer to this has changed over the last thirty years or so. When I was a kid, chemist's was invariably the term I heard, though pharmacy wouldn't have been incorrect. But recently, I don't think I've heard anyone say chemist's: it's always been pharmacy.
I've never heard a Brit say drugstore. Although we have a chain called Superdrug, I distinctly recall thinking the name was weird and slightly creepy when I first encountered it. I still do when I think about it, though of course one gets used to the anomaly.
I can't say how localised or typical this experience is. I'm from middle class southern Britain, but your kilometerage might vary.
The most common (and one that would be understood by 99.9% of people living in the UK), will call it a pharmacy.
Although, you'd get away with calling it a chemist.
People mentioning 'Boots' or 'superdrug' should note that that is a full shop, not just a pharmacy or chemist.
~Source Me (as an Englishman)
In order to clarify the terms chemist and pharmacy (although in the UK, people might/will interpret these differently (apparently)), i have included a 'comparison' for you:
A chemist is a scientist who develops drugs, and a pharmacy is a store that sells drugs. However, in Britain the word chemist is commonly used to denote a pharmacy