Wasn't that OTL between 1939 and 1941, with the exception of no Soviet troops in France/Denmark/Norway? Anyway, there were a few situation where Britain contemplated officially declaring war on the USSR - Operation Pike could do that, also Britain going through with its plan of sending troops to Finland during the Winter War. With the UK (and the rest of the allies) at war with both Germany and the USSR, the two are likely to end up coming closer.
But how much? The USSR doesn't have a common border with any of Britain's territories. It would have to either invade Turkey to get to the Middle East, Afghanistan to get to India. So it's ability to aid Germany would be somewhat limited, unless Germany would allow Soviet soldiers on its battlefields. This is't likely with Hitler in charge, as the man held a view that it was Germany and Germans that were meant to win the war, not its allies (especially sub-human slavs). Also, Hitler wanted to invade the USSR eventually, and he can't really wait with that until the west is totally defeated if that means having to postpone Barbarossa later than 1942. Stalin also was quite cautious about fighting Hitler's wars which he didn't need to, so it is almost impossible for the Red Army to land in north Africa or the Red Air Force to bomb London. He could try invading South Asia if pushed by Britain, but OTL talks between Molotov and Hitler show just how different German and Soviet goals are.
And there is the USA, which neither Germany or the USSR have a chance of defeating. It can't be crushed - the entire Axis and the USSR doesn't have the resources to do that. More likely the war would drag on for a few more years, the Soviets being bled out somewhere in Asia, the UK and USA being able to move troops and supplies there by sea that the USSR would be able to by land; the British Isles being safe thanks to US Army presence; Japan being combated as OTL; Hitler using Soviet weakness due to fighting the allies to betray Stalin and invade as OTL; D-Day happening and the Allies pushing further east once Germany capitulates.