"Back to Earth" is set nine years after "Only The Good...".
Kristine Kochanski is first presumed dead, and then revealed to have dumped Lister and stolen a Blue Midget.
Rimmer is no longer a human, but once again a hard-light hologram; with the memories of both the original Rimmer and the nanobot-resurrected version from Series VIII (suggesting a Mind Patch had been performed, although in the deleted scenes of Back in the Red it is revealed that the "new" Rimmer found the "old" Rimmer's diary in the wreckage of Starbug).
The Cat, Kryten, and Lister are the same, albeit older.
They are all back on Red Dwarf, which is no longer the nanobot-improved, streamlined version of Series VIII, but more resembles the bulky and decrepit version from earlier series.
Rimmer appears at least ten years older.
Holly is off-line, due to, apparently, Lister flooding him with ten years of bath water.
At the end of "Stoke Me A Clipper", Rimmer (hologram) gives Lister a disk containing his current personality. Lister accepts it and promptly forgets about it. Which explains why Lister didn’t load Rimmer up in "Blue".
At the end of "Only The Good..." Rimmer (human) eventually dies just before the ship is destroyed (being chased by the Grim Reaper).
The others (Cat, Kochanski, Kryten, and Lister) have repaired the device for traveling to mirror dimensions, and Kryten has blended the anti-virus with his nanobots, and go back to our dimension, just before the ship is destroyed.
Kryten introduces his nanobot-anti-virus to the ship, in just enough time to save it.
The ship is resurrected to full working order, but to slightly different designs than in the previous eight series. And fully restocked with Starbug and Blue Midget craft.
The crew (in the Starbug and Blue Midget fleet) by this point are now far far away from the mother ship.
Alternate speculation #1:Something takes out the resurrected crew, such as the Simulant Generals.
Kryten uninstalls the ship's default Holly, and installs the original Holly from Lister's watch.
Then they shut down unnecessary power, and load up the disk Rimmer gave Lister, and power up a new Rimmer (hologram) in hard light but a slight corruption makes him appear at least ten years older.
Alternate speculation #1: Rimmer (human) was killed by the microbes or the Grim Reaper and is resurrected as a hologram by Holly
Alternate speculation #2, based upon one of the unused endings of "Only the Good...": Rimmer (human) survives, but at some point in the 9-year interim is killed and resurrected as a hologram at whatever point in his life he died, hence the aging.
Alternate speculation #3: Rimmer (human) survives. Later on, Rimmer (hologram) returns and trades places with him.
Alternative speculation #4: Rimmer (human) dies. Rimmer (hologram) returns and has memories of Rimmer (human) uploaded into him.
Lister goes off the rails; Kochanski dumps him telling Kryten to tell Lister that she is dead. But she steals a Blue Midget and sets of in pursuit of the original crew or a gateway back to her own dimension.
The Rimmer that was seen from "Back to Earth" onwards seems to have the memories of both as he references the accident that the original had a part in ("The End") the chamelionic microbes ("Only the Good...") and the Despair Squid ("Back to Reality") which seems to indicate that he is a hologram of the resurrected version of Rimmer with the memories of the original inserted into him.
Rimmer's Memories were stored on a back up computer in Starbug mid section (as seen in Psirens when he is rebooted ) this section was detached before the front part exploded at the start of series VIII so would still be aboard Red Dwarf to be uploaded at any point.