The short answer would be no.

But first an aside: in the Bismarck case, Sweden 'technically' didn't pass the information to the British. They passed it to the Norwegians through their naval attaché Alfred Roscher-Lund, who in turn passed it to his friend, the British naval attaché Henry Denham. Naturally the Swedes were entirely aware that would happen, but politically it looks better. And in a few cases direct information occurred, which I'll get to.

As it were, Stockholm was a veritable den of spies during the war, with literally several thousand agents of foreign powers milling around. The German legation alone had 8-900 employees. The Norwegian resistance (Milorg) had offices there, the Soviet "trade delegation" had a veritable forest of antennas on their roof and were ultimately declared persona non grata. The Germans were pressuring to have the same done to Brits like Denham too, but Sweden resisted that.

The political situation was that the Swedish Social-Democrat led coalition government was not pro-German, even if parts of the officer corps were suspected of German sympathies (e.g. Supreme Commander Olof Thörnell). But the occupations of Denmark and Norway in 1940 had strongly turned public opinion against Germany and made Germany the most obvious and immediate military threat to the country. Thus, besides the neutrality policy, the Swedish government had no motive to help Germany, other than to the extent thought necessary to not provoke an invasion.

In 1940, in a story that deserves to be known better outside cryptology history circles, the Swedish "C-bureau" (military intelligence) had cracked the German Geheimschreiber the Siemens T52. Although the Enigma and its cracking is more famous (and more relevant to the outcome of the war), it was a significant achievement. The Enigma was a small portable machine, with 3-5 rotors used for radio communications, while the T52 is a behemoth with ten rotors, weighing over 100 kg and running off mains power. It was used primarily for cable communications, which the Swedes had access to, since telegraph/teletype communications from Germany to occupied Norway and their allies in Finland went over Sweden.

So at that point in time in 1941, the Swedish codebreakers were running at full steam, cracking hundreds of German communications per day. Since the strategic interest was aligned with what was now their biggest source of intelligence, that's where the vast majority of resources were being put.

So little or no effort was put into spying on the British. Strategically the British weren't considered a military threat, the resources were needed for the German stuff. Second to them, Russia was the greater concern. Even if they'd wanted to, there was fairly little British intelligence within their reach. There was relatively little radio traffic they could intercept and even when/if they did, they had not broken the British ciphers. Even if they'd been trying their best, it seems unlikely they'd be able to get their hands on much information the Germans wouldn't have been able to. Should Sweden have helped the Germans, they would've been best positioned to do so regarding the Soviet Baltic Fleet. But the Baltic Fleet had been trapped inside the Gulf of Finland by minefields and was largely out of the war. Sweden did share their knowledge of Russian navy communications (mainly of pre-war vintage) with Finland, and thereby indirectly with Germany. But with Finland being entirely focused on Russia, and with access to Soviet equipment seized in the Winter War, the Finns were well ahead of the Swedes regarding Russia by the time they allied with Germany. Which is why it was a great windfall for Swedish intelligence when a large amount of Finnish material was transferred to Sweden in Operation Stella Polaris, at the end of the war.

On an aside, it was Finnish radio intelligence intercepts against Britain, which were forwarded to Germany, that led to the sinking of Convoy PQ17 and PQ18. (Finland had a strip of Barents Sea coastline in Petsamo)

Out of the over 200,000 German transmissions decrypted by Sweden during the war, the most notable one (possibly the only one) that was intentionally revealed directly to the British government, was this - The British ambassador to Moscow Stafford Cripps was passing through Stockholm on his way back to London in the beginning of June 1941 (just a few weeks after Bismarck sank). He met with Cabinet Secretary Erik Boheman, and in their discussion expressed his opinion that Germany would not attack the Soviet Union, whereupon Boheman told him the invasion had already been scheduled to occur between the 20th and 25th that month.

Sources:

Svenska kryptobedrifter , Bengt Beckman (available in English translation as Codebreakers: Arne Beurling and the Swedish Crypto Program During World War II )

Om svensk underrättelsetjänst , Jan-Olof Grahn

Spioner i Stockholm under andra världskriget, Christer Jörgensen (article in the book Stockholm: historien om den stolta staden )