Did it not close due to diplomatic pressure and not war? I am not sure. But Japan continued to be very powerful on the mainland even as the Pacific was lost. I don't see how one year will change much.
The Burma Road was closed by the British for three months only starting July 1940 because of Japanese pressure. It reopened in October 1940.
Then in the Burma Campaign, the Japanese invasion of March-May 1942 closed it again. In late 1944 after two years of construction, the Ledo Road opened up a land route from India to China. Several months later in 1945, the British retook Rangoon which would open up the original Burma Road.
It seems like people are confused whether the original poster is asking if the Burma Road originally was built a year earlier (not possible), or if the Burma/Ledo Road was reopened a year earlier (late 1943 instead of late 1944).
The first (building of the original road) is not possible. But we can imagine various altenrative histories of the Burma Campaign where a land route to China was reopened earlier.
I think this will have several effects.
1) The tonnage going to the Chinese armies will skyrocket. This is mainly because most supplies flown over the hump did not go to the Chinese Army, but the US air force. Almost all additional tonnage can go to to the Chinese armies.
2) If the road reopens in 1943, Burma is presumably liberated to the extent that the Chinese armies in Burma (Force X and Force Y) are no longer needed. That means they can be deployed back to China.
3) All of this happens before the 1944 Ichigo Offensive which was Japan's last great victory. The Chinese defeat really did a lot to damage the credibility of the Nationalist government. With supplies coming into China, and the Forces X and Y in China, Ichigo probably fails. Japan may have some success, but China likely protects the Hunan rice bowl, saves the US airbases, and keeps control of the railway.
4) This places China in an ideal position in 1945. At the very least Canton is likely liberated, perhaps even Wuhan. This gives China some important victories before the war's end. This will do a lot to boost Nationalist prestige at home and abroad.
5) The Nationalists will be much better placed to move quickly in China to occupy Japanese held China before the Communists, making it easier for them to hold that territory.
I don't know if it's enough to cause the Nationalists to win the civil war. But they will enter it with more prestige and credibility, have an extra year worth of supplies to train and equip their soldiers (perhaps worth another 40 divisions), and be better placed.
Important questions are whether this causes the US to support the Nationalists more after the war instead of instituting an arms embargo and forcing Chiang into repeated ceasefires that benefit the Communists.
On the other hand, these changes are not likely to prevent the Red Army from marching into Manchuria and dumping tons of their supplies and captured Japanese arms to Mao's forces. To overcome this, Chiang and the Nationalists really need massive US support during the civil war period.
If these changes cause Chiang to not lose his best armies in Manchuria as happened IOTL, then Chiang may either win the civil war outright, or at least keep China south of the Great Wall or Yellow River.