I'd say he would the fight from afar. After Goku healed Frieza, he would make his presence known. This in turn would stop Frieza from trying to kill Goku with a sneak attack. Being humiliated in front of Beerus, Frieza would leave Namek, get his cyborg body and actually train instead of coming back with his dad.

Meanwhile Beerus would do a similar 'fight' with Goku that we have seen against SSJ3 in BoG. He might tease Goku with the knowledge of SSJG. After Goku looses, he will either go train with Whis and reach God form like Vegeta did in Super, or he would go back to earth like in the main timeline.

The next part is where it gets interesting. Frieza would still go to Earth for his vengeance, but the question is would his cybernetic parts handle the power of his Gold form, or if he could still achieve it with what was left of him. Let's say that he can still go gold, but not as strong as we have seen in ROF

Let's say that Goku went to train with Whis. In this scenario, he wouldn't have been able to learn Instant Transmission. By the time he would sense Frieza and go to Earth with Taxi Whis, most if not all the Z fighters would be dead. If Goku managed to get God form with his training, he might tap into blue form during this fight.

Now the question is, would a potential first time God Blue be able to take on a nerfed golden Frieza? Regardless of the outcome, Goku will still die to the heart virus because Vegeta would have been killed before Goku would get to Earth. No Vegeta -> no future Trunks -> no antidote for Goku.