Simple: after the reapers defeat people start asking why the warnings of commander Shepard where not heard, thus more and more planets are asking
for independance, thus creating a 3 way galaxy. On one side, you have council space, who is now weakened by the loss of major millitary
forces and political collapse. Among other things, the citadel was the heart of the economy, so the situation is bad everywhere and life conditions have dropped sinificantly.
Then you have the independants, who are planets who left council protection to join a new federation/ conflomerate lead by one of the new characters. There you can find Krogan who don't want to follow Wrex, Geth who don't want peace with the Quarians and Batarians that are trying to rebuild their empire.
Then you have the Krogans/Geths/Quarians who have made a very successfull union, where the krogan are the main millitary
forces, the Quarians provide ships, while the Geths provides Data and gear.
Tought all of thoses sides wish they could remain independant, there is a conspiracy going on and war is about to start soon. That's the setting.
Now what happened is that the council created their own Cerberus, I called it Aegis but you can name it whatever. It's basicly Cerberus, if cerberus where the good guys and where using every species instead of just one. That would be where Shepard team ended up assuming they survived, So you would have say, Miranda second in command, Garrus in the strike
force, Liara on the intel side (I think that now that the alliance and possibly the council knows who she is, she is far too dangerous to be left alone), and EDI has the base AI. Shepard would be in charge, assuming he survived.
From there I can kinda see two options : a) You are now the leader of Aegis strike
force wich would make a
game very reminicent of the previous one, and where you could try to live up to Shepard with some of the people you know, or

you could use previous setting to create a pirate character kinda like what Shepard would have been if he had never joined the millitary. Thus you would be an independant
force that would go around doing services
for various organisation, and you would have
your own cruiser, that would either be stolen or an older model but patched up with some crazy stolen tech, meaning that the ship could look old and still be awesome (kinda like the hyperion in sc2). The ship has to be bigger, if there is no ship it's not mass effect, if it's smaller than the normandy, it's a regression, if it's the same size, it's going to feel like it's a copy, so it has to be bigger.
Then you can bring all the characters you did'nt bring, Batarian tech guy (everyone
wants one), Turian biotic lady (I know there is one in Omega but still), Female human sniper( or ingenier), Krogan philosopher (just to break the stereotypes, he could be some shaman whose clan did not survive the war, and thus ended up teaming up with you), Salarian infiltrator (could have lost faith in the governement or something), Asari assasin( Could be lawful evil, in it
for the money
for exemple)...
Basicly you could be aligned with all the 3 factions above, or true neutral and you would get a different ending from there.