I had actually heavily considered writing a short story based around this AH.
Although based in the present day, I had worked out a rough timeline. The basics of it that the discovery of resources e.g. gold/oil had convinced Russia to keep Alaska.
This influx of resources helped to increase the fortunes of the Empire, creating a powerful Eastern side of the Empire. The 'Great Trans-Siberian Railway' is built earlier. An emerging middle class is seen to occur, ending serfdom.
Russia still goes to war with Japan in the early 20thC and ends up victorious, taking several Japanese territories.
Russia is much better prepared for WWI and emerges with Nicholas still as Tsar. However the wall street crash hits Russia hard. Following decades of unrest, further wars with Japan and a failed bloody invasion from the Nazis, the Monarchy is disposed in WWII with Joseph Stalin coming to power and creating the Soviet Union.
The Japanese do not attack Pearl Harbour, focusing on a full war with Russia. The US continue with their isolationist policy and do not become involved in WWII. The War carries on until 1946 when Britain finally manages to slowly liberate France and defeat the Nazis. Hitler commits suicide but only after launching four atomic bombs over Europe.
And that was about as far as I got timeline wise! Obviously it was rather under researched but it made good fun. Another aspect of it was Canada never truly becoming independent due to fear of Russian invasion, and it acting as a buffer zone between the US and Russia.
The modern day would have seen Alaska as a breakaway state, trying to claim independence and some form of international crisis between Russia and Britain. Probably involving nuclear weapons.