There was a George W. Nichols that was elected Major of the 6th "Southwest" Missouri Cavalry in September 1862, at the same time John Coffee was elected Colonel. He apparently left the regiment later that year and was succeeded by Moses W. Smith, about the same time Coffee was replaced as commander by G.W. Thompson.
Why he left and where he went I don't know. He could be the same soldier listed on the rolls of the 1st/3rd Cavalry as a private. There are also several other G.W. Nichols you'll find in the Missouri Soldier's database, including one who served in Benjamin Elliot's Cavalry. Elliot served in the same Brigade as the 6th, which makes this Nichols worth looking into as a possible match.
Grundy county is quite a bit further north than the counties most of the 6th's soldiers were recruited from, but I know of many members of the regiment that did come from northern Missouri counties, so you never know.
The MO Soldiers Database can be accessed using the following link:
Good luck with your research.