So:
Say October 1st 1939 - Britain & France declare war on the Soviet Union.
Almost immediately in Britian, the known leaders of the Communist party join Mosley and the BUF on the Isle of Wight.
If we say that activity on the Western Front remains as OTL, conscription, which was enabled on September 3rd 1939 continues apace. by June 1940 the British Army alone contained 1.65 Million men. In 1941 the Pre Barbarosa Soviet Army numbered 2.09 million men, roughly 1/3 more.
Final mobilisation was around 4.5 Million men for Britian in 1945, while the Soviets managed 30 million by the end of the war.
At this stage of the war though, as huge as the Soviet Army would become, what size Army could Stalin afford to commit to a campaign against the British?
If aside from the DOW, everything else is OTL, then the SU need to guard it's eastern borders against the Chinese Nationalists and/or the Japanese. Against the British & Canadians potentially just outside of Alaska, (another way to bring the Americans into the war - the Soviets attack an American ship thinking it's British or Canadian), They also allegedly had in excess of 2 million men stationed in the west, (this does not match with the pre war Army size figures given so I would revise the total Army size upwards to around 3m+ in an estimation).
I would actually suggest that rather than the Soviets actually moving south towards the Middle East or India, that in fact, they would send no army at all.
India had a number of anti-imperialist/Communist groups in circulation, so I would suggest that instead, covert support for such groups would be hugely increased in order to try and forment a revolution in India instead. Palestine and Egypt could end up in a similar situation.