It took nearly 50 years (Earth measurement) for the galaxy to fully recover from the Reaper War, tho in distant or unexplored sectors it's unknown if it even recovered.
The Geth were decimated, but not destroyed, by the Crucible as it destroyed the Reapers. They had figured out at the last minute how the Crucible had worked, and developed a shied against it. 212 Geth Platforms survived undamaged, and 67 were heavily damaged, but repairable. Exact numbers were never given to Shepard, but she was assured that the Geth would not be exterminated. Having worked tirelessly on the repairs of the mass relays, they were granted a spot on the new Council.
The Council itself changed drastically after the war. Volus, Hanar (Drell numbers were considered too few to give them their own seat, so the Hanar traded off with them, alternating between meetings), Elcor, Krogan, Quarians... all were given a seat and a vote therein. Newer species were first given an Embassy on the Citadel, but were not allowed to fully join the Council until they were able to fully integrate into the galactic society. Only the Batarians did not get a seat. But they didn't survive the war, either. Kar'Shan was never colonized.
The Krogan rejoiced the end of the Genophage after the relay to Tuchanka was repaired. Wrex ruled the Krogan well for 340 years afterwards, but towards the end of that time, had noticed that history was repeating itself. In order to forstall another Genophage, Wrex and several Salarians came up with a different solution. The exact solution was never released publicly, but everyone noticed that the Krogan birth rate dropped, averaging about once every 40-50 Earth years. For the first Krogan Councillor, Wrex chose Urdnot Bakara, who served admirably well for several centuries.
Aria returned to Omega, where she ruled with an Iron Fist for centuries. Upon her death of natural causes, the people of Omega gave her an honorable funeral, since Shepard's calming influence had caused her to defend them, ruthlessly, against several aggressors. She had not been loved, but well respected. Her death caused a power vacuum that totally destabilized the station. Civilians eventually fled the station again, and it was officially shut down. Merc groups occasionally use it as a base, but it never again rose as the capital of the Terminus.
Joker committed suicide 3 weeks after the war. He never recovered from the loss of EDI. Shepard, still hospitalized due to injuries, was unable to attend personally, tho some helpful Geth allowed her to watch it live.
Tali never returned to Rannoch alive, choosing to instead work alongside Garrus in rebuilding the galaxy. They eventually married, and adopted several children, both Turian and Quarian. Garrus, to his dismay, eventually became Primarch of Palavin. They were both buried on Rannoch, in a plot near the Monument that was erected at the site where Shepard forged peace between quarians and Geth. That peace never ended, as an attack on one of them was met with strong hostility from not only the other group, but also from the Council.
Javik eventually married Liara, and together they ran the Shadow Broker until his death. Liara had 10 children from Javik.
James Vega eventually gained N7 status himself and had a long and illustrius career in the Alliance, eventually becoming Fleet Admiral.
Jacob Taylor married Brynn about a year after the war.
The Lawson sisters eventually lost contact with Shepard, but reputedly had lived long and prosperous lives.
Samara returned to Ria, and kept watch over her until her death. Shepard's deception was discovered, but Ria was allowed to live, with Samara becoming the new head mother of the Monastary.
Kasumi literally disappeared shortly after the war, her fate unknown. Many said that she had retired to be with her late partner's Greybox, and others said that she was at Shepard's funeral, but nothing was ever proven.
Zaeed retired, rather firmly, after the war. He lived out his life in Rio.
The Systems Alliance, at Shepard's firm insistance (some called it bullying), formally pardoned General Williams. Her words were: "If I can be forgiven for taking extreme steps to protect this galaxy, then he bloody well can be forgiven for taking extreme steps in protecting one colony." The Turians also honored him posthumously, and the Williams family was grateful.
As for Shepard herself: She never fully recovered from her injuries in the war. Having developed PTSD, she retired from both the Military and the Spectres and married Kaidan 4 months after the Crucible was set off. Precisely nine months from the day of it's firing, she gave birth to a daughter which they named Crystal. She would have 3 more children by Kaidan, a boy named Jason, and a set of twins, which were named Christina and John, respectively. She died of several health issues on April 11, 2200, plunging the galaxy into mourning.