One part of the Michaelson-Morley experiment is not that light travels at variable speed, but that it travels through the luminiferous aether at a fixed speed and that the Earth was moving through the aether and that the aether might have some movement of its own.
If you assume that this is where the differences start, and that light travels at a given speed through the aether, which is stationary relative to the galaxy, then the main problem comes in the changing speed of light in varying directions (if the aether is not stationary relative to the galaxy, then that would farther complicate things and I won't talk about this).
The sun is currently orbiting around the center of the galaxy at a speed of about 220 km/s, and the earth is orbiting the sun at a speed of about 30 km/s. This would cause the speed of light to potentially vary by up to 500 km/s as measured by scientists on Earth. This may seem like a huge number, but the speed of light is almost 300,000 km/s, so the difference is not that big.
500 km/s change is actually less than the error that scientists had had when calculating the speed of light for almost 200 years after it was discovered that light had a finite speed.
Furthermore, relativity could no longer be the case. It requires that light always travel at a constant speed, so this would cause superluminal travel to no longer be that big of a deal. It simply becomes a matter of accelerating yourself enough, and you don't have to worry about your inertial mass increasing, or time dilation, or length contraction.
Also, red-shift and blue-shift are now caused by an object's movement through the luminiferous aether. This would wreak havoc on astronomy as many astronomical observations depend upon it (such as the universe expanding). Christian Doppler relied on light not traveling through a moving medium in his original discovery of the Doppler Effect in 1842 (before the Michaelson-Morley experiment).
The lack of relativity would actually simplify making a GPS system because modern GPS systems have to adjust for relativistic time dilation.
This lack of space-time curvature also would make the inner solar system more stable because space-time curvature caused by relativity and the sun's mass could make Mercury's orbit unstable and cause it to get more eccentric, eventually being flung out of the solar system, colliding with another planet, or falling into the sun in the next few billions of years.
There are probably much more implications, but I don't think there is a way around the two assumptions that I made (light moves at always the same speed relative to the aether, as thought be the luminiferous aether theory and that the aether's wind is very little relative to the movement of our galaxy) without overly complicating things or destroying the entire universe.