Your experiment will measure the speed of light to be the same no matter what direction you do you it in. When we say the speed of light is constant we mean that every local experiment to measure the speed of light will find the same value. Even if we put our apparatus into a rocket and fire it off at 0.999c the experiment will still measure the same speed of light as it did when it was sitting on the Earth.
From a common sense perspective this seems silly. How can the measured speed of light be unaffected by the motion of the experiment? I don't know of any intuitive way to explain this, but it's a premise of special relativity, and all the weird effects like time dilation can be explained from the constancy of the speed of light.
I think the best way to explain the constancy of the speed of light is to explain it as a geometirical property of the universe. I went into some detail about this in the answer to Special Relativity Second Postulate. You might be interested to have a look at this.
