The Winter War has become pretty notorious as one of the best examples of a Pyrrhic victory in recent history. When I read about the war one thing that really confuses me was that the Stalin decided to initiate it in late November of 1939. Most basic military strategy I am aware of says that wars, especially offensive ones, should almost always be avoided in the winter, the course of the Winter War seems to bear that out with the Soviets suffering terrible casualties and initially failing to achieve their aims in part because of the difficulties of the winter climate and terrain. With that in mind, why didn't they just wait longer into the spring of 1940 before launching the war when conditions would presumably be more favorable to them? Was there a wider political reason like the position of the Nazis and Western Allies that made them antsy to deal with Finland as quickly as possible? Were there other climate considerations I'm not taking into account like Spring rains and thaw making it too muddy for vehicles and armor?
While we cant know for sure what was in the heads of the Soviet leadership at the time, especially Stalin himself, most evidence points to the Soviets thinking that the Finnish campaign would be a quick affair and a "fait accompli" to present to the world.
The Soviet infantry division was at the time a heavy formation with two artillery regiments, often (and always in Finland) a tank battalion attached with a large degree of motorisation.
The Soviet leadership probably believed that the war would only last a couple of weeks. They also seem to have believed that substantial parts of the Finnish society would be loyal to the puppet government under the exile communist Kuusinen set up in Terijoki.
The Soviet plan was for their heavy divisions to punch through any resistance the Finns could put up and by a kind of "shock and awe" demoralise the Finnish army and over-run them with tanks supported by massive amounts of artillery, both light and heavy.
The Soviets seem to have over-estimated the abilities of the Red Army and seriously underestimated the Finnish army - this was the case even in 1941, when the Soviets seemed to have assumed that one of their divisions would be superior to a German one due to its larger and heavier artillery component (this was not the case, to say the least).
The weather during the first weeks of the campaign was also relatively mild - early December saw temperatures between +2° C and -6° C, with a snow cover of about 10cm. It was not until the third week in December that temperatures dropped below -15° C consistently, and not until mid-January that the famous super-cold temperatures of the Winter War came around, with temperatures from -33° C to -49° C were reported over Finland.
So,f or the original Soviet plan, the weather cooperated quite well with little snow and mild temperatures - it was just that Finnish resolve and military competence combined with the lack of the same in the Red Army threw a gigantic wrench into the machinery of the plan the Soviet leadership had come up with.
They were in a hurry, but couldn't have started earlier.
They thought that it would be a walkover, so the winter shouldn't have been a big problem.
The urgency was due to the international situation at the time. Essentially, the answer to your question
Was there a wider political reason like the position of the Nazis and Western Allies that made them antsy to deal with Finland as quickly as possible?
is a big YES. The Soviet Union was cautious in its expansionism. The Molotov-Ribbentrop Pact, which placed Finland, Estonia and Latvia in the Soviet sphere of influence (which the Soviets interpreted as German permission for conquest) had only been signed on 23rd August, 1939. Before this, nothing could be done safely enough. At that time, the Soviet Union was fighting Japan at Khalkhin Gol/Nomonhan, with the heaviest fighting in July and August. A ceasefire with Japan was agreed on 15th September, to come into effect on the 16th. On 17th September, the Soviet Union launched their invasion of Poland, to pick up their share of the spoils as agreed in the Molotov-Ribbentrop Pact. This was followed by further talks with Germany, leading to the German-Soviet Treaty of Friendship, Cooperation and Demarcation of 28th September 1939, which adjusted the German-Soviet border in Poland, and assigned Lithuania to the Soviet sphere of influence. There was no peace treaty ending the Soviet invasion of Poland, since the Soviet Union treated the Polish government as no longer existing and the Polish government did not declare war on the Soviet Union (and neither did Britain and France, who didn't want to add more enemies while fighting Germany). The Soviet invasion operations ended 6th October, when the last organised Polish unit still fighting surrendered.
Negotiations were ongoing with the Baltic states, leading to agreements of "military cooperation", allowing Soviet baees and troops, with Estonia (28th September), Latvia (5th October), and Lithuania (10th October). The Soviet Union then waited until mid-1940 before using its military presence to take over the Baltic states.
The negotiations that would lead to the Winter War began on 5th October, with the Soviet Union demanding that the border on the Karelian Isthmus be moved to the west (i.e., away from Leningrad), Finnish fortifications on the Isthmus be destroyed, and a Soviet base at Hanko, with compensation in the form of territory from Soviet Karelia north of the Isthmus being given to Finland. It isn't clear exactly what the Soviet intentions towards Finland were. It is possible that all they wanted was the border adjustment in the Isthmus to protect Leningrad and the base at Hanko, and if they had gotten those, Finland would have been left alone when the Baltic states were forcibly absorbed into the Soviet Union. There are many who, quite reasonably, doubt that Soviet intentions were so benign, and that Finland would have faced the same fate as the Baltic states in 1940.
Whatever the Soviet intentions, they preferred to gain their demands by negotiation, which didn't risk war with France and Britain. The troops that would be required for the invasion (if it turned out that an invasion was required) started to move into place in October - an invasion was not possible until November. if there was war, it would be a winter war unless the Soviet Union was willing to wait until spring. Could they wait? The key question was how long France, Britain, and Germany would be at war. If there was a quick peace agreement, Germany's greed might turn to the east, with the Soviet Union as their next target - it would not be a safe time to invade Finland. If Scandinavia responded to the war with mutual defence treaties, relations between the Soviet Union and Finland would become more complicated, even if Finland didn't join such a treaty. If Scandinavia joined the war, things would become even more complicated (regardless of which side they joined on). The risks with waiting until spring were that (a) WW2 might not last that long, and (b) WW2 might grow to big for Finland to be safely invaded. Since the Soviet Union though that invasion would lead to quick victory, once it was clear that negotiations weren't going anywhere (at least quickly enough for the Soviets), and the invasion forces were ready, they invaded Finland on 30th November.
France and Britain promised aid. If Finland would accept French and British troops, they could be sent through Norway and Sweden, preferably with Norwegian and Swedish permission. In no such permission was given, then they could still send troops. However, substantial British and French forces would need to stay in Norway and Sweden to keep communications safe, and to stop German interference. If this would cut off Scandinavian resources (e.g., Swedish iron ore) from Germany, then that would simply be a nice side effect of helping Finland in her gallant fight. Finland's fear was that such intervention would expand their war, not end it. Most British and French troops would be deployed in Norway and Sweden, against Germany, and the forces actually reaching Finland might be too small. Finland's response was to continue negotiating with Britain and France for such intervention, and using those negotiations and the threat of France and Britain joining the Winter War to encourage the Soviet Union to offer acceptable peace terms. They succeeded - just as the Soviet Union had been in a hurry to begin the war, they were also in a hurry to end it.