Right, but light before the beams were activated was traveling at normal speed. The beams don't go off until 2186;in Andromeda it'd have to be the year 2,502,186 before the beams even began to light up the Milky Way.
Actually wait, no I've got that wrong. Hold on, do we know what speed the light from the beams was moving?
MY BRAAAAIN
This is always the problem with thinking too hard about FTL - because it violates relativity you're gonna run into paradoxes. For instance you can always find an inertial frame of reference where the order of events is reversed (i.e. if i leave earth and travel to the citadel at FTL, someone else will see me arrive before i've left). Then you have to deal with time travel as well.
As for the beams, they're not only crossing the galaxy at FTL speeds, but also exploding at each relay at FTL speeds, so they're clearly not made of ordinary "light", which then begs the question of how we can see them, since our eyes can only detect photons. Only explanation I've got here is that the Space Magic Radiation interacts with the normal matter in the galaxy (e.g. ISM) and causes it to light up in RBG (literally, pick one), which we can then see. This means that in order to see all the explosions at once, as happens in the cutscene, the camera must be positioned at the same distance from all of them (e.g. directly above the center of the milky way), although even then there would be major time differences since not all relays are the same distance from the center. So again.. space magic. And yes, the light from the explosions wouldn't reach Andromeda for 2+Myr, so if we arrive via FTL before then and tell the Andromedans about how our galaxy is fucked and the Reapers are everywhere, they'll just be like 'wtf you on about mate'.