The only chance for the Warsaw Uprising to succeed was the Red Army. Let's say Germans fail to counterattack at Radzymin in late July 1944 and the Soviets really reach Warsaw. The Home Army starts the urprising, manages to capture 1 or 2 bridges and clear the way for the Soviets across the Vistula river. German troops panic and evacuate Warsaw.
What would happen next. The most possible scenario would be the same what happened in Vilnius (
https://en.wikipedia.org/wiki/Operation_Ostra_Brama) or Lvov (
https://en.wikipedia.org/wiki/Lwów_Uprising ) - front troops coopoerate, after successful battle NKVD arrests Home Army leaders and disarms the soldiers. OTOH, here situation would be a little different - Warsaw was the Polish capital after all, and it would have been harder too keep the uprising quiet. However, the Soviets would probably do what they did in Vilnius and then claim that people of Warsaw indeed fought against the Germans and helped the Red Army, but did it under the leadership of Polish communists, while the Home Army mostly stood and watched. However, disarming a much bigger number of Polish insurgents would have lasted longer.
The reaction of western allies wopuld be, IMHO, similar to OTL. They needed the Soviets more than they needed the Poles.
The most optimistic (while unrealistic) scenario would be that Poles manage to get Polish Prime Minister from London to Warsaw and proclaim that Polish legitimate government is back in Polish capital. The allies grudgingly put some pressure on Stalin and eventually the The Provisional Government of National Unity is established with politicians from London and Polish communists (as it happened in 1945 IOTL). The communist position would be rather strong, since they did have an army (Berling's Army) and full support of USSR. Non-communist faction, OTOH, would have hundreds of thousands of Home Army soldiers, brave, but usually poorly armed (no tanks, no artillery, no air force). The Home Army would try to organize regular units, but the Soviets/ Polish communists would obtruct it in every possible way, promoting pro-Sovier Berling's Army. Eventually, Home Army would be more or less neutralized, its commanders arrested under trumped charges or murdered, and soldiers forced to join the Berling's Army.