There are three plausible results of the fighting in the Dyer Ridge sector on the 20th. One, the Confederates could have routed this line and sent the Union units fleeing back to Chattanooga as actually happened. Two, the fighting could have reached a stalemate with the Union holding the line but leaving both sides in no condition for further fighting. Three, the Union could have achieved an enormous success and driven the Confederates across Chickamauga Creek. Of these three it seems that option two is the most likely outcome. Why? In the Civil War there are few instances of an attacking force driving an enemy from a prepared position, especially when the defenders have ample artillery. In major battles there are only four instances where an attacker pierced an entrenched line; the Mule Shoe at Spotsylvania, the ending of the siege of Petersburg, the storming of the heights at Missionary Ridge and at Nashville. Two of these instances involve an army on the verge of collapse at the end of the war, Petersburg and Nashville. At Missionary Ridge the Confederates did not have good fields of fire for their artillery. At the Mule Shoe the artillery had been withdrawn the night before and the attack was begun under the cover of dark. These three different failings of the defenders were not things that the Union would have had to contend with along Dyer Ridge at Chickamauga.
Another situation to consider is that Wood might have been able to hold on long enough to receive ample reinforcements and there might have never been any fighting on Dyer Ridge. Longstreet's column was five brigades deep. The first line was Fulton's and McNair's brigades followed by Gregg's brigade. The third line consisted of Robertson's and Sheffield's brigades with Benning's brigade behind Sheffield. The final line consisted of Humphrey's and Kershaw's brigades. Assuming that this weight of numbers would not have been enough alone to drive Wood from the Brotherton Field position it would have fallen to the troops on Longstreet's left to carry the position. There were two divisions on Longstreet's left; Preston's division of Gracie, Kelly and Trigg's brigades and Hindman's division of Anderson, Deas and Manigault's brigades. These men would have fought Davis' division of Carlin and Martin's brigades, Sheridan's division of Lytle, Laiboldt and Walworth's brigades and Wilder's mounted infantry brigade. Van Cleve's division of Samuel Beatty, Barnes and Dick's brigades was in motion as the breakthrough occurred. They could have quickly reinforced Wood's position if Rosecrans determined to make his stand at Brotherton Field.
If Van Cleve's division only aided Wood's position that would leave six Union brigades to defend against six Confederate brigades. Considering that the defender usually was the victor in Civil War battles (especially when attacked by an equal number of troops) the Union would likely be able to hold this position. Another factor in their favor is that Wilder's brigade was equipped with Spencer rifles and this increased firepower would have helped them greatly.
The Union defenders did hold the Confederates back at Lytle Hill and Dyer Ridge for about 15-20 minutes so it is quite likely that Wood could have held Longstreet about that long at Brotherton Field which would have given Rosecrans the option of either holding onto that position tenaciously or withdrawing to Dyer Ridge. A disadvantage of holding the line at Brotherton Field is that it would have been difficult to bring artillery to the front due to the close proximity of the Confederates and the likelihood that they would have been able to kill many artillerists and their horses.