The Western Allies had been very sympathetic towards the Finnish cause during the Winter War, and there were certainly no enthusiasm for declaring war against the Finns during the Continuation War.

At Soviet insistience, the British declared war on Finland on the 6th of December 1941. However, the British warfare against Finland was limited to supplying the Soviets with lend-lease aid to Murmansk (much of what was used against the Finns since that front was close to the supply at Murmansk) and the occasional assistance with squadrons from the RAF.

The 151. Wing (consisting of Hawker Hurricanes of the reinforces 81. and 134. Squadrons) was deployed in the Kola peninsula in September and October 1941, before handing their fighters over to the Soviets (after having trained them flying and maintaining them) and returning to Britain.

USA never declared war on Finland, and the Consulate in Helsingfors/Helsinki remained open until 16th of July 1942. Diplomatic ties were not fully severed until the 30th of June 1944.

It is interesting to note that the British demanded that the Finnish cease military operations on the 3rd of December 1941 - and that Finland formally replied that it would cease offensive operations (which did not satisfy British demands, and a declaration of war was issued on the 6th).

The US issued a demand that the Finns cease military operations and retreat to the 1939 (not the 1941 one) border, essentially recognising the Finnish attempt to retake land lost in the Winter War.

The Finns publically rejected this demand, but President Ryti privately and in secret asked Mannerheim to halt any advances on the 5th of November 1941.

Mannerheim contacted General Hjalmar Siilasvuo, commander of the III. Corps that flanked the German XXXVI. Corps in Finnish Lappland and asked him to halt his advance. Just before this the Germans had reinforced III. Corps as they saw its advance as the one with the best potential to break the Murmansk railroad.

Here's a map of the situation in October-November. Finnish III. Corps are with the 6. SS Gebirgs-division 'Nord' (that has been sent to reinforce them).

Here's another map with the Finnish III. Corps included.

It is possible that III. Corps halted their advance because of stiffening Soviet resistance. It is also possible that the Soviets could have stopped them with further reinforcements along the railroad should they have been able to advance further. It is also possible that they could have advanced and stopped due to Western Allied pressure.

The general consensus as I understand it is that Finland had prepared for a short war and a quick collapse of the Soviet Union, When this did not happen by mid-October 1941, they realised that they might very well lose and the unofficial Western Allied pressure was to leave the Murmansk railroad alone, so that the British (and eventually the Americans) could supply the Soviets through it, or they would take an unfavourable stance towards Finland once the war was over.

To summarise, the Western Allies did very little to harm Finland during the war, and only Britain and the Commonwealth declared war, although they carried out very few hostile acts beyond supplying the Soviets through Murmansk. The US even recognised (in an offhand way) Finland's right to its 1939 borders. Both applied pressure to Finland to not sever the Murmansk railroad, something which Finland did heed.