Let's try to understand first the setting we are in. The sentence:
the fact that the speed of light is a universal constant and has not changed
seems to imply that the laws of physics and the universal constants they depend on remain invariant under this change of sizes. So what we are dealing with is a doubling of the distances between all physical objects without modifying the universal laws and constants of physics.
One missing element that's very important in this question is what happens to speeds of objects after the change. As the question doesn't say anything about them, let's assume they are unchanged.
Notice that almost everything in our world has a characteristic size completely determined by universal physical laws. Consider, for example, the size of atoms: Coulomb's potential together with quantum mechanics is sufficient to specify the (size of the) region the electrons are in. Distances between atoms in a solid or a liquid are also determined this way by electromagnetic force. In smaller distances, the nuclei, and even protons and neutrons have a universal size.
So if these sizes were doubled, nothing would hold together: solids and liquids would become gases, electrons and nuclei would be free, etc. If we were the same size as before, so that we can watch this happen, we wouldn't need any measurement of the speed of light to notice it.
Perhaps a more familiar example is the trajectory of the Earth. Its current orbit satisfies that the gravitational force of the Sun acting over it is approximately the one necessary for circular motion. If the distance is doubled, this force is divided by four, and with the same velocity, the earth woudn't stay in its circular motion around the Sun.
Now, to answer the question about the test with the speed of light:
If somehow you manage to do a measurement under this conditions of the time light takes to go from one object to another whose distance previous to the change you know, the result will be that light takes now the double of the time it took before, and thus you will know that distances have doubled.
About the last question involving time, I understand you are not referring just to the speed of clocks but to the speed of everything.
If the speed of every object is halved at the same time distances are doubled, the scenario will change a bit with respect to the one in which just distances were doubled, but the main qualitative conclusions will be the same. Observe, for example, that half the speed is not enough for the Earth to stay in circular motion; instead, $1/\sqrt{2}$ of the speed is needed.
