By October 15, however, the rains had intensified, Rasputitsa was in full swing, the Germans were stuck in the mud, and Moscow was safe despite Zhukov reporting he could muster only 90,000 soldiers to man the next line of defense on the Mozhaisk-Kaluga axis.
Most speculative discussion has asked "what if" the rains hadn't come or had come less severely. Zetterling and Frankson argue in The Drive on Moscow, 1941 that AGC would have captured Moscow but for the mud. Their argument is convincing but taking Moscow in 1941 doesn't mean holding it - a likely outcome of the usual speculation is a Stalingrad-style debacle in the capital once the burgeoning reserve armies deploy in December and drive the Ostheer back. Hitler would have insisted on holding the city and the result would likely have been a fiasco.
Instead of "later Rasputitsa," what comes from an earlier Rasputitsa?
We know that the post-encirclement force ratios in front of Moscow dramatically favored the Ostheer, yet the Germans were unable to execute deep operational penetrations in October's second half despite their overwhelming superiority. They pushed the Red Army back, yes, and continued to capture thousands of demoralized Soviet soldiers at the tactical level, but the lightning advances of early October were finished.
Given that dynamic, it's likely that, under similar weather/road contions in early October, but with far lower force superiority, AGC would have failed to encircle and destroy the bulk of the Soviet forces.
There'd also be impact elsewhere: AGS executed an encirclement on the Nogai Steppe north of the Azov Sea, capturing 106,000 PoW's by October 11 and wiping out two Soviet field armies (9th and 18th). To accomplish this feat, Kleist's panzers had executed an extremely rapid advance from their positions north of the Dniepr's great bend, where they had finished the Battle of Kiev in late September. The rapidity of the advance so surprised the Soviets that when the Germans took Mariupol on the Azov there remained fully functional rolling mills and thousands of tons of steel products. https://en.wikipedia.org/wiki/Battle_of_the_Sea_of_Azov (article contains accurate citations).
Had Rasptutitsa arrived 10-12 days earlier - in the last few days of September or first few of October - AGS's October victories would have been less dramatic as well. Probably Rundstedt pushes Southwest Front back and takes the Crimea, but there's no big Kessel and progress is less deep. Probably the Red Army retains all of the Donets region and Mariupol, and AGS never enters Rostov.
Between Taifun and the Azov encirclement, the Germans claimed 780,000 PoW's. While the Red Army acknowledged fewer losses, its record keeping in this period had collapsed and analyses like Zetterling's convincingly show that the German claims are accurate.
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What are the strategic implications of nixing the Ostheer's early-October Kessels on account of weather? IMO they're potentially war- and history-changing:
Combat power implications
Glantz in When Titans Clashed reports that the German-facing Soviet Fronts reported 2,2mil men for duty on November 1, 1941 - the low point of the war. Adding back 780,000 constitutes a 35% force delta. Given the slowness of German progress in OTL November '41, when the advances resumed on the hardened ground, a 35% force delta should be sufficient to stop the Germans entirely everywhere on the front.
For many reasons, that 35% numerical delta is a floor for the actual combat power delta conferred by avoiding the October encirclemts. Most importantly, an encircled army loses high-value personnel - artillerists, supply specialists, staff officers, even generals - that "normal combat" does not highly attrit. These men had higher education and were often younger than the high-churn rifle infantry. See Dunn, Stalin's Keys to Victory: The Rebirth of the Red Army, p.21. Despite the Red Army's recovery in December '41, a glaring weakness was a surfeit of infantry masses and a lack of trained specialists and capable officers. As a result, its blows were clumsy and uncoordinated - a situation remedied somewhat in this ATL.[see David Stahel's new book Retreat from Moscow for good discussion of the Ostheer's exploitation of RKKA's clumsiness during that winter].
Morale implications
The October 15-16 mass panic in Moscow and large-scale evacuation is well-documented. Troop morale suffered too: during October 23-27, XXIV PzCorps captured 2,481 prisoners, mostly from 6th Guards Rifle division. There was no operational encirclement so these were all tactical surrenders. That a supposedily elite unit had such a high surrender rate is an indication that Soviet morale was battered.
Production and population implications
Land conquered by Germany after later-September was some of the most important taken during the war. AGS took critical cities of the better part of the Donets basin - Donetsk, Sloviansk, Kramatorsk, Horlivka - as well as Kharkov, Mariupol, Berdiansk, Izyum. The newly-occupied agricultural regions of Eastern Ukraine were a boon to Germany and their absence nearly catastrophic for SU. [See Hunger and War for discussion of mass starvation in SU's industrial cities in '42-'43.] AGC took or damaged important industrial centers of the Moscow region, e.g. Tula's environs (coal mines), Kaluga, and Kalinnin. Coal production in the Moscow region didn't recover until '43.
Some portion of the male population of the lost territories remained behind and was lost to the Red Army, which later inducted millions of soldiers as it liberated terrioty.
With significantly stronger forces all across its front, the Red Army's Winter Offensives would have been significantly more successful than OTL. As it was, the Ostheer barely averted dramatic defeats involving the encirclement and capture of large units. See Stahel's Retreat from Moscow. Half of 4th Army was cut off from the main body for weeks and only rescued by Model's first famous "fireman" action as an army commander. Throughout December and January, large gaps in AGC's front persisted, which didn't lead to encirclement only because AGC found sufficient forces to fix front by the barest of margins. As the Red Army is significantly stronger, more experienced, and with better morale than ATL, it seems likely that some large-scale encirclement and loss would have befallen Ostheer.
Red Army would also be advancing from points further west with greater strength, resulting in recapture of more land than OTL. A reopening of Leningrad via land, AGC being pushed back to Smolensk or farther, and AGS being back on the Dniepr is feasible.
That conclusion to the winter campaign implies more Soviet resources in '42 and therefore a stronger Red Army.
Meanwhile, if the soldiers not captured during Moscow and Azov battles in this ATL are traded at ~3-1 rates with the Germans, then the Ostheer has ~300k fewer men for the '42 campaign.
Increasing the Red Army's strength by, say, 10% in '42 and decreasing the Ostheer's by 12% is a sizable shift in the balance of power and probably forecloses any Blau-like offensive success by Germany. That's especially true if the altered force ratios prevent the pre-Blau, Spring '42 disasters (Izyum and Trappenjagd), saving another ~400k Soviet PoW's.
Seeing Germany decisively thwarted and pushed back, Germany's Allies would likely begin looking towards the exits. Less Romanian fuel would flow, fewer Axis satellite divisions would join Ostheer in '42.
With that setup, the Red Army can start its Dniepr-Carpathian drive in summer '42 rather than latter '43.
Because Germany never takes the Donbas, Eastern Ukraine, and North Caucasus, by '43 is the SU is materially and numerically significantly stronger than OTL, while Germany is weaker.
The Red Army could be in Berlin by the end of '43, certainly by early '44.
What would the Wallied response be?
They'd have to launch an invasion of Europe by '43 to beat the Red Army to the Rhine.
Maybe they invade in '42, seeing Germany on its heels.
The war might be over a 18 months earlier.