What is pretty certain is that none of the allies care to admit in their history any role played by English gold or arms. I don't recall reading a word about all of the money and arms that Austria received from England in any of the books from an Austrian perspective. No doubt the same would be true of any Russian orientated history.
I'm not aware of any history tracking what happened to any of the million or so English muskets once they were shipped off to the receiving countries. One of the very few that I've ever seen was that a British ship with 40,000 muskets intended for Hungarian Landwehr (?) was captured by the French when they took the port of Trieste in 1809.
I recall reading that soon after Russia was attacked by N. that the Baltic ports were filled with hundreds of British ships carrying arms and supplies. What role did these supplies play? There is, as far as I am aware of, nothing at all written about it. But, just because something doesn't show up in the history books doesn't mean it didn't happen or play an important role. The only real records we seem to have are the English account books.
There was no decisive battle until Leipzig 1813 and I have to believe that at least some Russian infantrymen would by then have been carrying some of the couple hundreds of thousands muskets shipped from England. If not, what the heck did the Russian do with all of those muskets?
For certain, Alexander had access to a million or so English Pounds (deposited in London at his request as I recall) that helped him pay the bills. War ain't cheap and a bankrupt country usually cannot afford to go to war.
While it may be hard to give a clear picture of what role British gold, credit, and arms played in the eventual victory of the Russians over the French; it is pretty easy to assess what would have happened without it. Russia didn't have the financial means to field an army and move into central Europe in 1813 without British aid. At least that what's they stated to the British.
So, it might be more correct to say that English aid "enabled" the Russians to carry on with the war into Germany and eventually Paris; which turned out to be decisive.