If the Soviet Union joined the Axis

Would the Axis have won if the Soviet Union became an Axis power?

No; the Allies would likely defeat Japan first and then turn against the Soviet Union, with the extended war probably bringing another nuclear attack.

Even Soviet entry into the Axis is not expected to produce an Axis victory. The Allies would first concentrate on defeating Japan and then turn against the Soviet Union. A longer war against this enlarged coalition would make another nuclear attack likely. The target is left unspecified, but the sequence still ends with the Allies dismantling the opposing powers rather than accepting their victory.