There are a few things to clarify:
the speed of light is c in vacuum when measured locally
the speed of light can be different if you measure it from far away (from a different gravitational field), this is the Shapiro effect
You are asking basically, if the speed of light would be different, let's say half the current one, c/2:
locally, nothing would change. As per SR, and GR, we are in the same gravitational field, and if the speed of light would be c/2, we would not really recognize the difference, to us, this would be the normal speed.
measured from far away, it is very possible to measure a speed for light slower then c, as per the Shapiro effect, let's say you try to measure the speed of light ass it passes next to a black hole. What would this world next to the BH look like from here (Earth)? Every clock next to the BH would seem to tick slower (GR time dilation) relative to our clocks here on Earth. So basically, if you set the speed of light to be c/2, our world would look (from a far away observer's view) much slower.
Now you might be asking about something different. If your question is, whether if the speed of light (locally) would just change to a arbitrarily slow speed, and this change would be relative to everything else (meaning all other fundamental constants of physics would stay the same), then the answer is I believe it is not possible.
All the fundamental constants we know about are somehow connected to the speed of light (propagation of information). All matter that builds up our world, is built up by quarks, and gluons. Gluons are massless too, and propagate at the same speed c.
If you change c, you change the propagation of gluons, thus change all the binding energy that builds up matter. If that changes, then the change of speed of light (and thus gluons) will relatively cancel out (mass and energy equality), and our world will become again the same (or seem to be the same) as it is now.