In the original timeline, Sweden stayed neutrality after the German airborne takeover of Oslo and the German Wehrmacht soldiers invasion of entire Norway in a gradual forwarding pace in northward direction. The reason Sweden stayed neutrality was the importance of the transported iron ore that need the fuelling for the German transportation and railway fuel and war machinery. The Germans needed iron ore imported from Sweden to maintained self-sufficient for the vehicles fuels during the wintertime. This was result of German invasion of Norway in an effort to secure much of captured naval bases to prevent British naval fleet sailing at Norway who want a fully security to protect the western Scandinavian region.
But let's say Sweden heard the German attacking of Denmark that marked the preparedness German takeover of Norway, the neighboring friendly country at the east with Sweden. On April 11, Swedish Prime Minister Axel Pehrsson-Bramstorp announced on the radio and telegram throughout Sweden that the Swedish government has declared war on Germany as an effort to help and protect slacked Norwegian military forces around Norway by directing major efforts to maintained northern Norway as remaining Allied territories. This would meant the fully denial of German workers and military forces to have an access for Swedish iron ore to maintain Hitler's economy full war production.
What could be the impact if Sweden joined Allied to declared war against Axis powers Germany in an effort to highly prevented entire Norway country collapsed by the outnumbered superior Nazi German troops? Would British troops being sent to Sweden to protect the importance economic industrial interests at Stockholm?
First of all, Per-Albin Hansson was the Prime Minister of Sweden in 1940, not Axel Pehrsson-Bramstorp.
Secondly, the scenario is very unlikely. Sweden, for all its common history with and sympathy for Finland did not declare war on the Soviet Union when it attacked Finland - despite the Soviet ability to strike at Sweden being severely limited - both in trade and supply and in military terms - compared to Germany. That Sweden would declare war against Germany when it was unwilling to do so agains the Soviets (with Russia being the historical arch-enemy of Sweden, not Germany) is borderline ASB.
Germany supplied almost all of Sweden's need of coal, and Sweden was way more dependent on German coal than Germany ever was on Swedish iron ore. Swedish iron ore was convenient for Germany, since the high ore content and ability to ship it by sea and then river barge to Rurh made a ton of steel made from Swedish ore cost about half in resources and money compared to steel made from domestic German ore, even when facturing in the transport costs. Sweden would not go to war with Germany without having secured a supply of coal first. Sweden needed coal and coke for its own steel production, to run its electrical generation plants (Swedish railroads were electrified and without coal the trains would stop since electricity would become scarce). Swedish hospitals also ran their own backup power and disinfection and laundry steam generation on coal. Not even speaking of producer gas in the cities running on water-coke-gas.
Sweden's army at this time was defensively focused and lacked the ability to quickly and decisively attack and knock out German forces - in some cases, it was downright bad. However, the Germans used their C-team in Norway (no troops outside the 3. Gebirgs-division had any mortars, the mean training time was 90 days and a week or two's preparation for the invasion).
The Germans lack the ability to invade Sweden at the same time as Norway and Denmark, and Scandinavia was a secondary front for them - they will be unwilling to dedicate a lot of their air force from the front and the planend great offensive in the west, but they can certainly commit enough to overpower the almost non-existant Norwegian and small Swedish air forces.
Sweden had on the 9th of April 100 000 men along the border with Finland, which had been "topped" with the most and the best equipment of the army as well as trained during the Winter War. These were however in the process of demobilising and on their way home when Denmark and Norway were invaded. The aid sent to Finland and the "topping" of that Army Corps (2 divisions) had drained the army stocks, and there was a LOT of confusion during the Swedish mobilisation - which was aided by the fact that the Germans demanded that Sweden do not mobilise and Sweden calling it a "readiness increase" instead, which made some depots withold things like ammunition and uniforms that were earmarked for a mobilisation before the confusion was cleared.
However, Sweden knew that Germany was not coming for them in April 1940 - Swedish intelligence, a combination of diplomatic sources, military espionage and contacts with civilians was pretty good and warned Norway four times before the invasion - in the 31st of March, on the 6th of April and twice on the 8th of April.
The Swedish Supreme Commander, noting the German buildup suggested a Swedish partial mobilisation on the 4th of April, to go into effect on the 6th, but was denied, partially since the government knew the Germans were not coming for Sweden.
There is a way to make this happen, or at least make Sweden treat Norway like it treated Finland - declare itself non-belligrent rather than neutral and actively send aid and volunteers.
Sweden had a debate on defence policy and foreign policy in the 30s, with the "either-or" faction faving the "traditionalist" faction. The "either-or" faction, in the military haded by Colonel (later General and Supreme Commander) Helge Jung and Major General Axel Rappe and in foreign policy by Foreign Minister Rickard Sandler argued that Sweden either needed an army at the 1914 level in order to defend itself properly, or it needed to form defensive treaties with its neighbours to come to each other's aid if they were attacked as defined by the League of Nations.
The Swedish military discussed defence cooperations with Finland in 1938. However, the question was a sensetive one in Finland and Mannerheim sent Colonel Airo, one of his closest associates and a very hardline Finnish nationalist to head the Finnish delegation. Finland was in the midst of a conflict regarding language where Finnish-speakers and nationalists resented the traditonal Swedish-speaking upper class and their power and wealth. Sending Airo seemed like a good move by Mannerheim, since he probably considered the negotiations a done deal, since both nations wanted the same thing - a defensive pact and cooperation, and any deal negotiated by the known nationalist Airo would probably be accepted by other nationalists (who viewed the Swedish-speaking upper class' connections with Sweden with suspicion). However, Airo bungled the negotiations - he probably believed the Swedish military would never get the support of the politicians for any kind of deal and that the negotiations were a waste of time. So he insisted on not speaking Swedish (despite being fluent), so the negotiations had to be held in French, acted arrogantly and piled demands on top of each other and told the Swedish representatives that they were there to discuss what Sweden could do for Finland and not the other way around, so the negotiations ended in a failure.
This pretty much gutted the "either-or" faction in the Swedish military and when Sweden formed its grand coalition government and appointed a Supreme Commander, Thörnell, a known traditionalist got the position as Supreme Commander and Sandler was replaced by the German-friendly and staunchly neutral Christian Günther as Foreign Minister.
If you go back to 1936 and have the "either-or" faction in Sweden more successful, and Finland being more receptive of it, perhaps Mannerheim controlling Airo closer and Norway preparing a tiny bit more and being more receptive of any kind of defensive treaty, then you could get something like this scenario, or close to it. You also need the Norwegians to take the threat seriously in some way. The Norwegians considered it impossible for the Germans to attack them over a Royal Navy-controlled North Sea and believed that friendly relations with Britain and Sweden was all they needed to remain safe and did absolutely nothing to protect themselves despite multiple warnings. You won't see Sweden defend Norway if the Norwegians are not defending themselves. Case in point - during the first three weeks of the Winter War, Swedes were not very interested in helping, until it became clear the Finns were fighting like mad to defend themselves, then a huge surge of sympathy rushed through Swedish society and aid streamed to Finland.
However, you would need for the Germans to not take these deals seriously, as they know they do not have the capacity to fight Sweden and Norway at the same time before they have knocked France out of the war - if they take any kind of defensive treaty seriously they won't attack as OTL, as the German attack was reliant on Norwegian lack of a response, Swedish neutrality, audacity and a lot of luck.
