In brief: the relationship between Finland and "the West" after WW2 was complicated, but improved into "friends with benefits" territory by 1970s.
The Soviet threat and a desire to regain the areas lost in the Winter War had driven Finland to ally with Germany from 1941 until 19th Sept 1944, when a ceasefire was signed between Finland and Soviet Union. Politically, Finland remained a representative democracy. At the time, the Allied Powers generally had some understanding towards Finland's precarious position, as evidenced by e.g. the fact that Finland was specifically discussed as an exception to the 1943 "Casablanca Declaration" that required the Axis countries to accept unconditional surrender. Finland was also originally supposed to be spared from paying war reparations, but on insistence of Stalin this was later changed.
Again, on Stalin's insistence, Great Britain declared war on Finland on 6th Dec 1941 and even conducted some air attacks against the port of Petsamo. The United States, however, retained diplomatic relations. A September 1944 U.S. Department of State memorandum "Treatment of Finland" spelled out the U.S. policy towards immediate post-war Finland: The Government of the United States considered that Finland had been a victim of aggression, and had been compelled to fight in 1941. Therefore Finland, unlike authoritarian satellites of Germany, would have the right to form its own post-war government. (My emphasis.) This was of quite some importance since U.S. was the only government Stalin listened to, because of the continuing importance of Lend-Lease aid.
This U.S. lenience did not prevent the Soviets from imposing punitive war reparations and demanding the Finns to go to war in order to drive the remaining German troops from Finnish Lapland.
Once the Second World War was over, Finland's position seemed again precarious at best. Finland was officially neutral country, but as the relations between the West and the Soviet Union cooled in the immediate post-war years, Finland was generally written off as another country about to fall within Communist sphere of influence. Great Britain, for example, was initially reluctant to supply Finland with coal, as this was seen to benefit only the Soviet bloc; however, rebuilding Britain required vast quantities of lumber and Finland was one of the most important suppliers. This gave the Finns some leverage in trade and in foreign relations towards Great Britain.
However, gradually the U.S. and UK governments realised that Finland was not inevitably going communist, and began to improve trade and cultural relations in an attempt to counter Soviet influence in Finland. A turning point of sorts was reached in 1948, when Finland's disorganised Communist party could not even attempt a Czechoslovak-style takeover - despite widespread fears in Finland that the Communists were about to do so. (It's still disputed whether there even were plans to do so; historians generally think there was a lot of booze-fuelled tough talk but no real plans for a coup d'etat.) The Soviets had made clear to the Finnish Communists that they would not intervene on their behalf: the home-grown Communists had to be able to take power themselves, and only then the Red Army could be "invited" to secure their power.
Instead of taking over Finland, the Soviet Union pressured it to sign a so-called "Agreement of Friendship, Cooperation, and Mutual Assistance." The gist of the agreement was that Finland promised to defend its territory and prevent its use by any third party against the Soviet Union; were it incapable of doing this on its own, it would request necessary help from the Soviet Union. Originally the agreement would have amounted to a military alliance in all but name, but the Finnish negotiators were able to modify the wording so that military "cooperation" would only happen after mutual consultations that agreed that there was a threat of attack from "Germany or her allies" (meaning the Western bloc).
Such an agreement was advantageous to the Soviets, because it avoided the need to occupy Finland in peacetime - a prospect the Soviet foreign minister Molotov likened to a "bleeding pustule," given well demonstrated Finnish will to resist any such attempts. Furthermore, the agreement meant that Finland remained a neutral territory and a valuable conduit of Western technology and know-how to the Soviet Union: this was to prove useful to the Soviets in the later years.
As for Finland, the agreement fundamentally clarified the relationship with the Eastern neighbour and eased the worst fears of imminent occupation. From then on, the relationship game Finland played with the Soviet Union and the West began to stabilise. For the next decades, Finland's relations with the West were characterised by the Finnish desire not to slide into the Soviet bloc; that desire was tempered by the fear that overtly courting the West would provoke a counter-reaction from the East.
Generally speaking, as the years passed Finland's relationship with the West improved. Finland would have been accepted to the Western "clubs" (e.g. the European Free Trade Area) fairly quickly; the reason it stayed out for a long time were the aforementioned fears of Soviet response. During this time, the fears of the Eastern neighbour were used heavily in the domestic politics of Finland, resulting to - among other things - self-censorship of topics and issues that might antagonise the Soviets. This process of sucking up to a larger neighbour's unstated wishes is generally known as "Finlandization," a term much resented in Finland, even though the allegations are not without substance.
A major win for Finnish diplomacy was the OSCE conference in Helsinki in 1975: among other things, it cemented the acceptance of Finland's neutrality within the West, although the official Soviet acknowledgement (arguably the more important) had to wait until 1989 visit by Mihail Gorbachev. However, this was at the end of the Cold War, and in the 1980s Finland was still bound by the FCMA agreement. What would have happened in the event of war between the two power blocs is a topic of perennial speculation in Finland. Officially, Finland would have been required to defend its neutrality (a difficult proposition given several bomber routes to the Soviet Union crossed Finland's airspace); unofficially, the only scenarios ever taken seriously by the top military brass concerned those where an invasion comes from the East.
SOURCES
Zetterberg (ed.) (1987) Suomen historian pikkujättiläinen. Helsinki: WSOY.
Rautkallio, H. (Ed.). (2014). Suomen sotakorvaukset. Helsinki: Paasilinna.
Seppinen, J. (2008). Vaaran vuodet? Suomen selviytymisstrategia 1944-1950. Helsinki: Minerva.
What do you mean by 'the west'? USA and Nato?