First, lets look at the text from the ["Agreement of Mutual Assistance between the United Kingdom and Poland, Aug. 25, 1939"](https://en.wikisource.org/wiki/Agreement_of_Mutual_Assistance_between_the_United_Kingdom_and_Poland-London_(1939)).
Although the phrase "Should one of the Contracting Parties become engaged in hostilities with a European Power in consequence of aggression by the latter against that Contracting Party, the other Contracting Party will at once give the Contracting Party engaged in hostilities all the support and assistance in its power" would seem to be straightforward, it really isn't! There was also the Secret Protocol which stated that "European Power" was little more than a euphemism for Germany, and that if another power invaded "the Contracting Parties will consult together on the measures to be taken in common."
So yeah, there was a big, glaring caveat, and while the invasion on Sept. 1st saw Britain fulfil what was a clearly stated obligation, on Sept. 17th, Britain was happy to stretch that language to the breaking point, no matter how much they might not have liked Soviet actions. And why wouldn't they? There was definitely belief that Soviet-German cooperation wouldn't last and they would eventually turn on each other, and a declaration of war would jeopardize that, forcing the two into closer cooperation. Poland was doomed, and Britain knew she had no chance to actually save her from this initial invasion, so what would be gained by adding another belligerent to the war? When Sir William Seeds, the British Ambassador in Moscow, was asked his opinion, he responded "I do not myself see what advantage war with the Soviet Union would be to us, though it would please me personally to declare it on Mr Molotov."
So anyways, on the morning of the 17th, the Polish Ambassador visited the Foreign Office on what he really already knew was a futile mission, understanding that the British Government would (publically at least) state they had considered 'other European power' to be Italy, even though Poland, and perhaps any reasonable person, would have seen the USSR as the next biggest threat. The Cabinet met the next day to consider the matter, and prefered to do an ostrich act than play the lion, deciding simply to formally protest Soviet actions to register their "horror and indignation", and maintaining "complete confidence that on the conclusion of the war Poland would be restored."
Poland of course protested, communicating to Lord Halifax, the Foreign Secretary, that "the Polish Government reserve the right to invoke the obligation of its allies arising out of the treaties now in force", but was succinctly rebuffed, which Halifax emphasising the British interpretation of the treaty, namely that they had no obligation to do anything other than consider the matter, meant "we are free to take our own decision and to decide whether to declare war on the USSR or not."
So within only a day or two, the Soviet action was a fait accompli and if anything, the British government now went on a PR campaign to try and justify its lack of action. In October, after Poland had fallen, Lord Halifax addressed the House of Lords to note:
It is perhaps, as a matter of historical interest, worth recalling that the action of the Soviet Government has been to advance the boundary to what was substantially the boundary recommended at the time of the Versailles Conference by the noble Marquess who used to lead the House, Lord Curzon, who was then Foreign Secretary.
Churchill, at the time First Lord of the Admiralty, characterized Soviet action as stemming from "cold-self interest" but nevertheless saw them as a future partner against Germany who shouldn't be unnecessarily antagonized for no benefit:
I believe Russia will always act as she thinks her own interests demand, and I cannot believe she would think her interests served by a German victory followed by a German domination of Europe.
Privately, he counseled Chamberlain that it was a 'favorable development' for British interests, and in a public declaration a few weeks after the Soviet incursion, he went even further to declare that their action "was clearly necessary for the safety of Russia against the Nazi menace". Pragmatic about Poland and the Soviet sphere since even before the war, when he was perceiving the potential for it, he was under no illusions about the importance of the USSR as an eventual partner in the war against Hitler, and wasn't eager to upset future events.
Additionally, although not a member of the government, it is also worth mentioning MP Lloyd George's editorial in the Sunday Express entitled "What is Stalin Up To?" published in late September, which essentially justified the Soviet movement and pushed their own explanation that it was intended as a humanitarian action to protect the people who lived in eastern Poland (In the words of the Soviets, "The Soviet Government also cannot view with indifference the fact that kindred Ukrainian White Russian people, who live on Polish territory and who are at the mercy of fate, should be left defenceless"). When Ambassador Raczyński attempted to have a refutation published in the Times, he was rebuffed and had to arrange for private printing of it.
The sum of it is that, to quote Doerr:
[F]rom the signing of the Nazi-Soviet Pact to the beginning of the Winter War [...] British policy-makers were driven by a profound ambivalence. On the one hand, a deep and abiding mistrust of the Soviets is easy to find. On the other, the British remained acutely aware of the grave situation they faced, and of the overriding need to retain a connection to Moscow. Such ambivalence can be found at the heart of all aspects of British-Soviet relations during this period.
It was only upon the invasion of Finland that British lawmakers began to find themselves unable to justify their inaction in the face of Soviet aggression. The onset of the Winter War was simply much harder to explain away with magic handwaving though, and in the weeks leading up to the Soviet invasion, the British were discussing how they would need to potentially react to such an eventuality. There was fear that if the Soviet did so, it would be prelude to a push further into Scandinavia, possibly threatening Norway. Some within the government went so far as to suggest that the UK should immediately declare war if such an attack came, although it was the minority view - "ideological circles" in Lord Halifax's words. A telegram from the British Ambassador to Finland on Oct. 21 to that effect was deemed impractical within the Foreign Office since "the Cabinet presumably still take the line that we cannot afford to break with Russia and thus turn her into an ally of Germany" as one commentator noted. This was backed up further by reports from the military that they were in no state to be able to lend military assistance.
When the invasion came, obviously, war was not declared, despite significant outcry from the British public over the Soviet action. The same caution as before applied, but it was a lot harder to explain away what the Soviets were doing this time around. The British provided supplies, and there were some volunteers, but it wasn't until March that they finally felt they had no choice but to give into the public pressure and intervene. But even then, it was supposed to be a quite limited action, with the main intent to protect further incursion into Scandinavia, not just by the Soviets, but by Nazi Germany as well. The main thrust of the planned force was to occupy northern Norway and Sweden in order to prevent Swedish iron ore from falling into Nazi hands, and potential bombing of oil fields in the Caucasus, which at the time was being exported to Germany. The actual military assistance to the Finns would have been a single brigade placed in the far North, far from the key southern region where any actual impact would be felt. Obviously it all came to naught when Finland fell before it happened. This was just fine with the British as they had been incredibly uncomfortable with going through with the plan. It still was a bad blow to Soviet-Anglo relations for the next year though, and had at least some on Stalin's reluctance to trust British reports in early 1941 that Barbarossa was imminent.
In the end of course, the UK didn't necessarily make the wrong choice (for them. Poland and Finland definitely got screwed). I don't want to deal with counterfactuals, but the possibilities for how war would have progressed if the UK and France had entered open hostilities with the USSR certainly don't seem to improve the odds for the Allies. The travails of Poland through the war, and beyond, are really another topic, but suffice to say that the UK continued to ignore what was staring it in the face, and continue in its (public) belief that Poland would be restored, although in part we can say that it was a continuance of their "we need to not piss off the USSR" policy, and when it was clear they were mistaken, it was too late and there was nothing to be done anyways, leaving us with the sad irony of the war, that it was launched on the casus belli of maintaining Polish independence, and in the end, Poland would spend a half-century behind the Iron Curtain.
Doerr, Paul W. "'Frigid but Unprovocative': British Policy towards the USSR from the Nazi-Soviet Pact to the Winter War, 1939." Journal of Contemporary History 36, no. 3 (2001): 423-39.
Hastings, Max. Winston's War: Churchill 1940-45 Knopf Doubleday, 2010
Kochanski, Halik. The Eagle Unbowed: Poland and the Poles in the Second World War Harvard U. Press, 2012
Prazmowska, Anita. Britain, Poland and the Eastern Front, 1939 Cambridge U. Press, 1987
(Also don't miss this from /u/kieslowskifan)
The first thing to realize is that the full truth of the Secret Protocols only came out after 1945 and it was considered Nazi propaganda by the USSR and fellow travelers during the Cold War. While in truth the USSR, post-Purges, was left isolated and went for a pro-Nazi foreign policy, the UK and France of 1939 had no way to prove that Hitler and Stalin were so cold-blooded as to be able to veer from their lavish denunciations of each other to negotiating partitions of Europe in a totalitarian Tilsit.
As far as they could see, in 1939, the USSR invaded after the Nazis broke Poland and carefully used the Curzon Line proposed in 1920-1 as their stopping point precisely because it was a British line meant to avoid provoking the British and French too far. That was the picture in purely military terms.
Germany starts a war in Poland and deliberately provokes events. Poland, expected to hold out longer, folds in two weeks in real terms, and the Soviets come in only after it's broken and then only to seize territory on a line proposed by the British in the first place in lieu of the existing post-Treaty of Riga frontiers. The Soviets looked 'the hyena to the German lion', not a conniving regime that waited just long enough to seem more opportunistic than it ever really was.
On top of this, it should be noted that contrary to the usual stereotypes and to their relative passivity against the Germans that both the British and French leadership of 1939 literally proposed a hypothetical strike against Baku. Churchill, contrary to his stereotype did not see bombing Baku as worth the potential risks, and much of his decision-making stemmed from the logic that a recreation of the coalition of 1914 would be worthwhile. The logic behind this prior to the collapse of France was that the French would not fold in six weeks in an unexpected and unforeseen triumph of fascist arms. When France did, that threw the existing logic cockeyed, and when the British were driven out of Europe a third time in Greece in 1940 left the USSR as of June 1941-September 1943 the only Allied power fighting a European war in Europe, and tying down the vast legions that otherwise would have turned to face Anglo-American armies.
At that point what the USSR did in the first years of the war meant much less than the fear of facing the vast legions of the East in Italy or France, so pragmatism at Poland's expense became the order of the day. The defeats in Norway, France, and Greece tied the democracies' hands. Two German divisions and a much bigger number of Italian troops ran rings around larger and more well-equipped British troops in North Africa for most of a year while the USSR was fighting much larger battles in Europe proper and became in a perfect position to get whatever it wanted on the terms it wanted in terms of postwar gains as it became clear the Nazis were better at murdering civilians than waging war.
In an alternate universe where the narrow victory at Sedan becomes a somewhat broader margin of victory for Western arms, an Axis-Soviet War would not end near the way the historical conflict did on terms so neatly convenient for the USSR. The losses of 1939-40 and being driven out of Europe three times in a year did much to tilt the war, and to load the geopolitical cards in Moscow's favor to a sense that nothing short of a brand new attempt at a 1917-style regime change imposed at the business end of a rifle was going to reverse it. And that included the gains of 1939 at Poland's expense and in the Baltic states, and against the Finns.