WI Yudenich takes Petrograd in October 1919

First, let's suppose some changes compared to OTL. Kolchak's attempts to prevent the independence of the Russian Empire's western borderlands (Finland and the Baltics in this case) have to be removed, so we'll say that the British are able to push this point more aggressively. Yudenich's government in the Northwest was prepared to offer independence in exchange for support on his Petrograd offensive. Without Kolchak threatening the Finns, there's a good chance that Mannerheim is allowed to open a front in Karelia against the Reds. If Estonian participation is able to pay for their independence, Estonian support would probably also be more intensive. I don't know if better Finnish and Estonian support could change the outcome of the campaign by itself, or if we need PoDs to keep the WVRA from fighting the Estonians, but that's not really the question I'm asking.

Petrograd was effectively the birthplace of the Russian Revolution going back to the 1905 rebellion. What kind of impact does the loss of the city have on the Red war effort at this point in time? The Reds were on the offensive against both Kolchak in the east and Denikin in the south at the time, so I don't see how they could really lose those unless they halted the offensives and transferred massive numbers of troops back to the Northwest Front. IOTL, the Red forces (7th Army) gathered to defend Petrograd were primarily from the Northwestern Front, which effectively collapsed as a result, and this was what enabled the Baltics and Poland to gain independence. Red reinforcements were also ordered in front the White Sea area, but not from the eastern and southern fronts (at least not before the White offensive stalled). Lenin was in Moscow at the time but Trotsky had gone to Petrograd to supervise the defense, so it's not inconceivable that he could have been killed or captured if the city fell. I don't think this would have had nearly as great an impact at this late stage in the war as it could have had in 1918.
 
The administration and Bolshevik state apparatus by this point was all relocated to Moscow and I believe there was a communiqué from Lenin advising abandoning Petrograd to Yudenich so it is certainly possible the Bolsheviks could have withdrawn from the city without fighting to the death over it. In this case Trotsky would almost certainly escape barring any unforeseen accidents, but it would be a blow for sure. The British could potentially garrison the city alongside Finnish and Russian troops and use it as a staging ground to strike south towards Novgorod and onto Moscow in coordination with Denikin’s march from the South. Given the sort of propaganda value the defeat of Yudenich was, the morale blow could be immense and cause defections in the Red ranks:

Very recently the British reactionaries, the imperialists, made their last stake on the capture of Petrograd. The bourgeoisie of the entire world, especially the Russian bourgeoisie, already had a foretaste of victory. But instead of victory they met with defeat at Petrograd.
Yudenich’s forces have been beaten and are retreating.
Comrades, workers and Red Army soldiers! Bend all your efforts! Keep on the heels of the retreating troops at all costs, crush them, do not allow them to rest for an hour, for a single minute. At this moment we can and must strike harder than ever in order to finish off the enemy.
Long live the Red Army that is defeating the tsarist generals, whiteguards and capitalists! Long live the inter- national Soviet Republic!

N. Lenin
November 5, 1919
Petrogradskaya Pravda No. 255, Published according to November 7, 1919 manuscript
 
The fundamental advantage that the Reds held throughout the war was not just that they had interior lines that allowed them to quickly move forces around, but that they were essentially defending the Russian heartland against invaders from the periphery. Even more valuable than the interior lines was the strategic depth that Moscow's location afforded. The White armies were so far from Moscow that they had no realistic chance of taking the city before the end of a hypothetical 1920 campaign. Petrograd is 400 miles away, Kolchak was struggling to reach Kazan on the Volga 450 miles away, and Denikin's southern offensive was launched from 400 miles south of Moscow. The late summer White offensives made good progress, especially in the south, but they had no ability at the time to mount anything more than a "strategic raid" towards the city, which is what Denikin's offensive seemed to take in form. Because he didn't have the support of the Russian population south of Moscow, Denikin was effectively relying on the shock of Moscow falling to destroy the Bolshevik war effort and let him secure the Volunteer Army's rear logistics area. Of course, the whole Black Army episode kept that from happening, but I can't tell if Denikin's offensive stalled before or after the effects of the Black Army's attack on the White logistics system became a problem.

If Yudenich is able to take Petrograd in late 1919, it certainly would be able to serve as a logistics base for a 1920 offensive against Moscow from the northwest, but then numbers become an issue. Even if the Finns, Estonians, and Baltic Germans work together to take Petrograd, any further action would be entirely on the backs of Yudenich's Russians. They would probably need at least 40 to 60 thousand troops for an offensive of that scale based on other action in the war, but they didn't number much over twenty thousand even after a number of unit defections early in the 1919 campaign.
 
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