I've read the story, I think it's a good story, but if you think it through, its plausibility is debatable.
I don't recall if the writer ever said exactly how big the robots were. I was picturing them as maybe 1 foot tall, i.e. much bigger than 1 mm. The closer they are to normal size, the less dramatic any scaling effects.
As others have noted, some things would be fairly obvious, like objects would fall too fast, water puddles would be too thick, the air would seem more viscous. You'd probably notice some square/cube effects, like hot objects would cool off too fast, and you could put more weight on a table or chair without it collapsing. If you had the equipment and expertise and measured the speed of light you could prove it definitively. I don't know how chemists determined the number of atoms in a mole, but if you could reproduce that you could prove it. Etc.
We'd have to ask how far the builders went to deal with such issues. Maybe they didn't give the robots real water but some chemical that behaves in a way that "looks right". (And of course if the robots drink it, they're programmed to think it tastes like water.) Presumably the robots are built so that their strength, speed, etc "feels right". The acceleration of gravity would be tough to work around as long as Tiny Town is on the surface of the Earth. But if you get most things to feel right, maybe nobody notices a few odd things, or if they do, they write it off.
Of course, the whole point of the story was that people just woke up one morning in their robot bodies in this artificial world. Presumably they had no reason to doubt that this was not just another day. So ... how odd would things have to be before you started sayng, "Hey, this is weird"? Sure, someone with scientific expertise who suspected he had been miniaturized like this could think of dozens of experiments to test the theory. But would your everyday experiences be enough to make you ask the question? If you were shrunk to 1 mm, I suspect they would be. But if shrunk to 1 foot? Maybe not. It's really hard to say.
Like if I woke up one day and noticed that, say, a book I dropped fell faster than it should, would my first thought be, "My consciousness must have been uploaded into a miniaturized robot!"? More likely I'd say, "Hey, that's weird. Is that normal? Is there something funny about this book? Oh well, anyway, got to grab some breakfast and get to work ..." Even if there were many things, I'd probably think, "Wow, I'm just really slow today" or "Maybe I should see a doctor" or whatever before I'd think of being turned into a miniature robot. And remember they only had one day to think about it before their brains were reset and they started the day over.
I suppose we could postulate that the robots are all programmed not to notice any discrepancies. They could be programmed to think that the speed at which objects fall is what they've always seen before and this is perfectly normal, etc. In which case the answer would become, It's impossible to tell, because you're programmed not to.
When I read the story I wondered: if the builders of Tiny Town had no moral qualms about uploading people's consciousness into these robots and using them for these experiments, why not just cordon off a real town of real people and do similar experiments? Indeed they wouldn't have to cordon it off, just run their different advertisements and observe what happens. Well, okay, with actual people they couldn't do the daily reset to get a controlled experiment. But it would seem a whole lot cheaper and easier. If they have the political pull that they can requisition the bodies of all the people killed in this explosion and upload their consciousness, surely they have the political pull to get whatever laws they need passed to let them run their advertising campaigns in one little town. Etc.