Sweden having a significant empire in the 19th Century would presumably have significant effects on Russia, Germany and perhaps Britain:
- Russia would not necessarily find a more muscular Sweden a threat by itself, but Sweden could potentially change the tide in any conflict between Russia and either Prussia/Germany, the Habsburgs or Ottomans - they would be defending more than one front, so might either lose any given war, or be more reluctant to enter it in the first place.
What you are talking about was a typical scenario of the mid-/late XVIII and early XIX: Sweden was attacking Russia when it was at war with the Ottomans (prior to WWI Russia was never at war with the Hapsburgs and only once, during the 7YW, with Prussia). This never was noticeably effective in the terms of distracting the Russian forces from the Ottoman theater and the whole schema culminated with a permanent loss of Finland. The cornerstone of the Swedish policy under the Bernadotte dynasty was a neutrality and the biggest "anti-Russian" act was a treaty (1850s) with Britain and 2nd Empire with a purpose to protect Sweden-Norway from a possible Russian demands of getting a stretch of coast along the
Varanger Fjord.
Having a bigger colonial empire overseas would not change anything in that equation.
- Sweden could potentially intervene in the German unification process in some way - or else a newly-formed Germany could see Sweden as a threat to be dealt with, whether diplomatically or militarily.
In OTL Sweden supported Denmark against the
Kingdom of Prussia in the
First War of Schleswig by placing Swedish and Norwegian troops in
cantonments in
Funen and North
Schleswig and was also one of the guarantors of the integrity of Denmark (the
London Protocol, 8 May 1852). But a bigger colonial empire would not seriously contribute to the Swedish military power even if simply because a huge difference in the sizes of population between Sweden-Norway and Germany (or even Prussia of the mid-XIX).
- Britain may have an interest in supporting this Swedish Empire, seeing Sweden as a useful counterweight either to Germany or Russia; alternatively the British may see Sweden as a rival in the North Sea and attempt to pump up the Danes a little bit, which in turn may affect Germany again.
In OTL at various times Britain was pro-/against Sweden. "Pro" was mostly at the time when there was a realistic fear of Peter I trying to convert Baltic Sea into the Russian lake by the strategic marriages (Holstein, Mecklenburg, Curland; for a while he had Russian troops occupying Mecklenburg to support his son-in-law against the estates) and, of course, when Sweden was fighting against Napoleon. In practical terms, effectiveness of "pro" was quite limited: during the GNW British squadron did close to nothing to support Swedish navy against the Russians or to attack Russian-held former Swedish provinces and during the Napoleonic Wars they could not prevent Swedes from being beaten or from Swedish Pomerania being occupied by the French. Neither could they do anything to prevent loss of Finland. So their practical value was buying of the Swedish iron and providing subsidies for fighting Napoleon.
By the time "Germany" appeared on the map AND the Brits started considering it as a threat (instead of supporting it), Sweden-Norway hardly could be a serious naval or military factor comparing to Britain and Germany so it could not serve as a meaningful counterweight to Germany or Russia and, anyway, between early XIX and 2009 neutrality was an official Swedish policy.