Reapers reside in dark space (aka, void between the galaxies).
With their regular FTL, even at 10 000 x LS, it would still take Reapers a while to reach the outer edge of the Milky Way (in all likelihood).
The Citadel was described in ME1 to act as a 'Giant Relay', which in all probability allows it to transport the Reaper fleet from Dark Space, to the Milky Way.
Regular Mass Relays inside the Milky Way likely don't have the range needed to reach beyond the galaxy into Dark Space (they function as part of the network inside the Galaxy as we saw in the game by connecting to each other - fairly short ranges on a galactic scale - the Citadel is the only largest relay that has a single purpose - bring the Reapers into [and probably once the cycle of harvesting is complete, out of] the galaxy and take out the seat of Galactic government).
Once Nazara (aka Sovereign) was stopped from using the Citadel to call forth the rest of the Reaper fleet in ME1, the Reapers probably started using their regular FTL engines to get inside the Milky Way (the Alpha Relay in the Batarian system from ME2 Arrival DLC) and simultaneously started using the Collectors to abduct Humans from their colonies (since Humanity - and Shepard in particular - was instrumental in stopping Nazara) so they can build a core of the new Reaper (made in Humanity's image).
From the Alpha Relay, the Reapers then proceeded to arrive at Earth and begin their invasion (since Humanity peaked their interests).
What was also discovered in ME1 is that a group of Protheans on Ilos managed to delay the Reaper invasion fleet when they discovered the signal on the Citadel and how to manipulate it.
We don't know by how much was the cycle delayed, but Vigil (VI on Ilos) estimated it may have taken Nazara centuries to discover why the signal was not activated... someone else on the internet mentioned the invasion was delayed by 2000 years.
But the earliest attempt (that we know of) dates back to the Rachni Wars.
So, given that the ME races of the cycle we know of (that came after the Protheans) did not really advance too far and were nowhere near the level of what the Protheans achieved.
This would suggest that if the Ilos scientists never managed to alter the Citadel signal, the invasion would have happened hundreds (or possibly 1000-2000) years before Humanity became interstellar civilization (which would in all likelihood not warrant the Reapers from harvesting Humanity - and I wonder, just how far would Humanity go in the tens of thousands of years - they would still be limited to ME technology [obviously], but its still interesting).
This may also indicate that the Reapers would be pretty far outside the Milky Way galaxy.
2000 years delay seems a bit much because the Reapers would take, what, less than 300 years of continuous travel to reach Andromeda galaxy (but as we know, ME ships and quite possibly the Reapers don't usually have the ability to sustain FTL indefinitely due to static charge buildup which needs to be purged - it IS possible the Reapers don't have this issue, but they were also stated to go into Dark Space after each invasion to conserve their energy reserves and also replenish them during each invasion cycle - and we know from ME Andromeda that the big ships used to travel to Andromeda had technology that recycled the buildup charge allowing for continuous FTL travel at their top speed - but we don't know if the Reapers ever developed this - the Andromeda initiative concerned itself with preserving existing species and reaching Andromeda... whereas the Reapers focused on the Milky Way - there's no indication they were interested in other galaxies).
Its a bit inconsistent... because if the Reapers are supposed to awake every 50000 years (via the Citadel signal which the Protheans altered - and if that's the case, what or whom prompted the Reapers to become active? It was suggested they leave only 1 Reaper behind to monitor the galaxy and initiate the cycle), and if they were fairly close, they would have used standard FTL to reach the galaxy and the Alpha Relay much sooner if they were only 20 000 Ly's away from the galaxy (which would take them only 2 years).
If they are NOT that close though, then its possible the Reapers were already on their way via regular FTL from the moment the initial invasion was delay by the Prothean scientists (whenever that happened) - which begs the question, who or what awoke them?
Is it automatic for them? I don't think so, since it was suggested in ME1 the Citadel signal was supposed to awake the Reapers (signal them) and them activate the Citadel relay to bring them inside the galaxy.
It is possible I suppose that since the Collectors were in the galaxy, they awoke the Reapers (and because of the battle of the citadel in ME1, Nazara alerted the Collectors), but couldn't bring them in (1 ship or maybe a handful at most vs an entire Citadel fleet of Asari and Turians would likely prevent that regardless of how powerful their weapons are - we know the Collector ships are not good in regards to protection of their own hull - susceptible to regular weapons it seems and limited shielding).
Anyway, some things are a bit evidently out of sync.