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.
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