It doesn't change that much, IMHO, at least in short terms. During the Operation "Tempest" ("Burza") a significant number of Home Army units helped the Red Army by attacking German rear. Cooperation between HA soldiers and Soviet front units usually went rather smoothly. And then NKVD units would come, arrest the Polish officers, disarm the rest and give them a choice - gulag or joining the Polish Army controlled by the communists. It happened in Vilnius (Operation "Ostra Brama"), Lvov and in Wolhynia.
OTOH situation in Warsaw might be a little different. All cases mentioned above took place in territories which had been given to USSR in Teheran, so western Allies treated them as internal Soviet affair. Warsaw was (and still is) the capital of Poland and it was supposed to remain so - after all Stalin had promised free and independent Poland.
However, in Stalin's vision of future Europe there was no place for such a state. So I think the scenario might be similar like in Vilnius or Lvov, it would have only taken somewhat longer. Cooperation at the beginning, perhaps even some help in organizing HA units into a regular force (so they are easier to find), joint operations. After a few months arrest (or murder) of the officers and forcing the ranks into communist Polish Army, the Red Army or sending them to gulags.
Another thing - an initially successful cooperation with the Soviets in Warsaw might encourage other HA units to try to cooperate with Soviets when they reach the rest of Poland. So the "cooperation" period might last a little longer longer.