First, you are asking about the electron's electric field, and we should distinguish between the near and far field.
The electrons' electric field's (near field) force is mediated by virtual photons, they are just a mathematical way to describe this force. These are not real photons and are not real particles.
Though, the electric field (near field) extends with the speed of light, so it is not instantaneous.
In your case, since the electrons are stationary, and are already there, they already have a near field extended (around a certain position in space), and if the two electrons are close enough, they will feel measurable repulsion as you suggest.
Now the reason they repel each other instantly is because the near field they create is already there, always, and cannot be turned off. This way you cannot test the speed of these interactions.
The way to test what you are asking for, is with electric charges (artificial, man-made), that can be turned off/turned on. Once you put two of these charges close enough, you can test, what you are asking, that is how fast will these near fields interact?
So your question is basically, after you turn on the charges, how soon will they interact and repel?
The answer is, after you turn on the charges, the near field (electric) extends in spherical shape in space with the speed of light. So depending on their distance, it might take a while until they feel the other one's near field.
Let's disregard that in your question you are talking about distances of 300 000 km, which is too far away for man-made electric charges to create a near field strong enough to feel measurable repulsion. Let's say that these charges are 300 000 km apart. Let's say that we have strong enough electric fields, that will create a near field strong enough so that it will have a measurable effect 300 000 km away on the other charge.
Now let's say that we can time the charges to turn on simultaneously at t0. The near fields of the charges will extend with speed c. The two fields will not meet half way, at 150 000 km distance, because of the correct comments and because two near fields will not effect each other, but they will obey the principle of superposition, they will add without interacting.
So the field of one electromagnet will reach the other electromagnet at 300 000 km, in 1 second, and will start repelling it.