As both flags cannot be located today I have no clue what the 20th Tennessee's flag looked like. They also got a special flag issued to them in March, 1863.
Their Augusta Depot 1864 flag might only have had the unit designation on it in white cotton cut and sewn letters.
Their First National is at the Tennessee State Museum today.
The flag of the 13th USCT, most likely their National Stars & Stripes flag by the way, was taken to Alabama supposedly and it has since dissappeared.
Sadly, the capture reports of the Battle of Nashville, like those for Franklin, are not loaded with details in terms of what flags were taken by what Union regiment exactly. Some have those details but most do not - and that makes it very hard for us today.
Tennessee State Museum