Wasn't the point of the operation to support the Soviet Offensive anyways? If it's delayed until some later Soviet Offensive, it does seem rather perfunctory at that point. I honestly don't see it changing much at all at a strategic or other meta level beyond thousands of lives being saved in the immediate time frame.
When it turns out the Soviet Union isn't leaving Poland though, I doubt the Home Army will be able to do much to prevent it. In fact they'll probably be told not to do anything by the Western Allies, at least until the Pacific War is sorted out but even then, the Home Army, as impressive as it was, couldn't oust the Germans on their own in late 1944, there's no way they could oust the Soviets from Warsaw, much less Poland, on its own. Chances are the Home Army and the resistance network behind it either withers away and the amount of active resistance it does try to pull off will be inversely proportional to how many are killed or imprisoned by the Soviet backed authorities post war IMHO.
Only positive I could potentially see is if the Home Army and other pro-Western elements somehow survive until the 1950's and maybe help moderate the Polish Government and Armed Forces so that when Hungary does its revolt in 1956, maybe Poland (and perhaps others) could join in on it and also tell the Soviets to fuck off. That's the best case anti-communist (asnd therefore optimistic) scenario I could see but I feel its unlikely.