The Brits invaded Norway to prevent a German invasion (not, btw, a good idea), not to fight the Russians.
The British were considering an invasion of Norway, but the Germans beat them to it. The actual British troops sent were in response to the German invasion.
There were a couple of different British plans for Scandinavia. One of them was, indeed, to send British forces into Norway, through Sweden, and into Finland to assist the Finns in the Winter War against the Russians. That fell through when Mannerheim sued for peace. The second, in the early spring of 1940, was for a pre-emptive strike through Norway (then neutral) and into Sweden (then neutral) to seize the Swedish iron ore fields, from which the Reich was getting the majority of the iron ore it needed.
Hitler didn't have much interest in invading Norway at first, but when it became clear that the British might take steps to cut off the Swedish ore, and block the Norwegian sea lanes, that stirred him into action, and as a result the plans for the invasions of Norway and Denmark were thrown together hastily. The actual German invasion of Norway caught the British by surprise, but they did respond quickly.
}:-)4