In Axis and Allies, Which Ally Should "Take Out" Norway.
There are some subtle benefits of owning Finland: 


To Russia, it represents another 2 IPCs a turn, which it can turn directly into infantry on the front: a valuable ability. The US and UK can obviously also use the money, but it takes longer for the units to reach the front (first built, then moved to Finland, then to Karelia). Also, both the US and UK have to move units via transports, which means that sometimes even if they have the 3 IPCs for an extra infantry they don't actually have the transport bandwidth to move it the following turn.
The US can build a factory in Finland, thus enabling it to pump 2 units a turn directly into Europe. I typically don't do this if the shuck-shuck is going well because the opportunity cost of a factory is too high. Although I've done it in a long game where I have too much money and the Axis isn't ready to concede--just to speed up the tanks I can put on the continent.
I don't perceive any real advantage of UK ownership--other than the money.




That said, I usually try to have the first country able take Finland and don't worry too much about who it is.

In general, this seems to be Russia. Assuming no ally can take it on turn 1 (by rule for Russia, for want of sufficient transports for UK and US), then turn 2 is the earliest. If Germany was clever, they rescued the tank and 2 inf with a feint on Karelia, making capture easy; if not, it's a prime target for the Russian tanks and the necessary infantry (no dead zone concerns assuming the German North Sea transport is gone). This attack doesn't leave Karelia too weakened because of their turn 2 infantry placement and UK and US fighters that can be moved over in turn 1.