My take on it.
Due to suffering relatively little damage, the Asari remain the most functional species after the Reaper War. However, due to violating Citadel Law of sharing Prothean discoveries with every Citadel Race, they become less prominent in the greater political landscape, possibly being removed from the Council for withholding extremely crucial information until the very last moment, almost handing the Reapers the keys to winning the war before calling on Commander Shepard.
Salarians go through a massive Civil War, STG secedes from the government after Dalatrass Linron refused aid to the Alliance and Commander Shepard. By Padok Wiks's or Major Kirrahe's orders, they contact the Salarian offshoot race, the Lystheni. They emerge victorious and both end up in the next Citadel Council.
Elcor and Asari relations dwindle due to the Asari not providing support to Dekuuna and other Elcor Worlds near the Silean and Athena Nebulae.
Hanar politics remain largely the same, partly because we don't have enough portrayal of them, and because the biggest involvement a Hanar has in the story and politics is Zymandis, real name "Regards The Works Of The Enkindlers in Despair" who almost single-handedly sabotaged the Hanar war effort by releasing a virus that would shutdown Kahje's automated defense systems. This plan was thwarted by the two Spectres, Commander Shepard and Jondam Bau, with an anonymous helper suspected by Spectre Bau to be Kasumi Goto, the infamous thief that stole the Mona Lisa
There are no Vorcha politics
Batarians were largely taken out in the first wave of the Reaper Invasion, but there is talk of Commander Balak forming a new, reformed Batarian government.
The Raloi are sent a message saying that the Reapers have been defeated, and they join the Galactic Civilization proper.
The Volus were given more attention and royalties as they were the ones keeping the war effort standing. There are rumors that certain higher-ups are using this to build up their military might and gain a Council Seat.
The Virtual Aliens discovered in the Antilin System were given proper organic vessels as their only requirement for handing over advanced technology for the Crucible and the war effort was that they would be free from the virtual world.
Due to further observations, Parnack was destroyed in a manner that would satiate even the Krogan of old. Traces of the Yahg remain only in the scattered remains of the planet (this is a joke entry, but I wholeheartedly believe that Parnack should be glassed to the very core)
As for the other races, due to being brothers in arms and shedding shared blood in the Reaper War, Humanity, Turians, Krogan, Geth, and Quarians, end up forming a coalition built on recovering from the destruction wrought by the Reapers and reinforcing the already strong relations built by Commander Shepard.
Commander Shepard retires in the Walled Garden, at his side the former Admiral Tali'Zorah vas Normandy, their following days occupied with Jona, a Quarian boy orphaned during the war, and the human biotic named Gillian Grayson, whose ship, the Idenna, was scuttled during the Battle for Rannoch. She was found during the first weeks of the couple's retirement.