Well, not exactly: the world is still made of atoms, like (we think) our world is. But for some reason, there was no Democritus or Leukipo in this world, and, on the contrary, the mainstream philosophers believe that matter is infinitely divisible.

So what consequences does that have for the development of philosophy and science? I am pretty sure that technological improvement to the levels of the Renaissance wouldn't be troubled. But from then on? How would they explain electricity, for instance? And would it be necessary to correctly understand electricity to use it in practice? (I fear we didn't correctly understand electrons at the time we were starting to use electricity.)

Or would it be possible to explain such things as electricity, and even nuclear fission/fusion, within the frame of a non-atomistic comprehension of the world?

As I understand it, your question is whether the absence of certain key people in our history, would lead to modern day difficulties in identifying and explaining currently known scientific phenomena.

The answer is that it wouldn't, for two reasons.

First:

Generally speaking, once enough knowledge becomes available for a discovery to be made, waiting for that discovery to be made is just a matter of time. Often several different individuals make the same discovery around the same time. As an example we have the invention of calculus, which was discovered by Newton and Gottfried Wilhelm Leibniz independently. Many discoveries follow this pattern. So if one key thinker was absent, someone else would have made the same discovery, theory, etc, and gotten the credit.

When we think of Newton as being the first person to discover and theorize about gravity, the key point is the "first" part.

Second:

Even if the starting point were different, scientists would eventually converge on the theory of the atom. At the end of the day, it doesn't really matter that ancient philosophers theorized this or that, since it's modern science that was used to deduce the existence of the atom.

Conclusion:

At worst you would get a somewhat delayed scientific timeline, but not by much.

It's kind of hard to say for sure, but I think AlexP's comment is on the right track: EM is still possible, but chemistry probably would have bumped into problems first. To elaborate, Maxwell's equations, which govern electromagnetism, are very much compatible with the idea that matter is a continuum. In fact, they're not just compatible, they're mathematically easier that way-- a continuous charge distribution works perfectly well within the analytic world of the equations, while handling a discrete distribution requires adding a whole bunch of tools like dirac delta functions that weren't put on a rigorous foundation until the mid 20th century. So EM is fine within your world. Since Maxwell's equations were the motivation for special relativity, it follows that SR and even GR could probably be developed within your world.

Chemistry, however, is a bit tougher to reconcile-- the idea of indivisible elements sticking together to form compounds is just so simple and intuitive, yet so good at predicting phenomena we see. However, you could probably fudge some kind of continuum version of classical chemistry together, it just would be a lot less intuitive and clunkier to use. Where you really start running into problems is quantum mechanics and the like, since quantization is a fundamental aspect of these theories anyway. As before, you might be able to cobble together some atrocious continuum explanation that does a reasonably good job at predicting phenomena, but the models needed would get more and more complicated, less useful, and less unifying, until someone finally had the epiphany that discretized matter was the way to go. The whole situation would likely play out kind of like Ptolemy's model of epicycles.

TL;DR-- Scientific advancement would halt at around where it was at the turn of the 20th century.

In order to achieve this you would need to restrict scientific inquiry.

The first problem you are going to run into is that just because one person didn't make an observation, doesn't mean that no one will ever make that observation. The real question is how are you going to keep people from discovering the truth and making the observations that will lead them to the conclusion.

In a narrative story the easiest way to do this is to add more important threats to survival. It's really hard for people to make observations about atoms if they are running away from bloodthirsty cannibals. It this case, these more important threats are going to shape the world more significantly than any lack of scientific understanding. Also, in this case there is no such thing as a "Mainstream Philosopher", and if there was he was most likely called "Dinner" for the cannibals.

I guess we can't really give you an answer unless you tell us how it came to be that no one has observed the many clues that lead us to modern atomic theory.

Yes, this is entirely possible.

People once thought that atoms were the smallest possible unit of matter.

They were wrong.

We now know that sub-atomic particles exist: quarks, muons, leptons, and so forth.

With our current technology level, we are unable to determine if there are yet smaller units or not. We have even theorized that there are, in string theory.

By using a different paradigm of thought, we can reach a great understanding of what we call science from a totally different point of view.

One such example in today's world is the divide between Eastern and Western approaches to medicine. Western science remains unable to account for some provable, repeatable treatments and methods of reaching a diagnosis used in Eastern medicine. This doesn't stop them from actually working.

Another example is how physics once proved that bumblebees can't fly. Yet they do. And eventually they found out that their wings become flexible at high speeds, which they never thought to test or account for and this threw off all their equations and assumptions.

After all, once upon a time nobody thought something could be in two places at once (super-positioning), or be able to talk across great distances (telephone), or see things happening on the other side of the world as it happened (live TV), or teleport (we have now teleported atoms), or fly (the Chinese put a man in the sky via a giant kite looking before the Wright brothers)... I could go on.

By approaching matter/energy theory from a viewpoint of overlapping fields of biased energy types, one can essentially replicate all of our modern advancements without ever admitting the existence of atoms.

After all, all atoms are is a collection of measurable and non-measurable qualities which overlap and interact with each other grouped proximally in a local space time matrix. In other words... an energy field in a constant state of flux, containing a multitude of varying densities, reactivity, and permeability. It is simply convenient and easy to think of them as discreet objects, even though they actually aren't.

Today's physics are already exploring this concept: do a few searches about the holographic universe concept to see some of what I'm referring to.