One of the first effects was actually quite bizarre: summer heat. Surprisingly, the weather got unspeakably hot across most of Europe from June to August during the first phase of the eruption. This may, of course, be attributed to the fact that not only SO2 and HF but also CO2 are released during such eruptions, but the true culprit remains a mystery. Of course, that heat wave also resulted in the formation of severe thunderstorms large enough to produce hail. In Europe specifically, this hail, which would have been similar to the hail that we modern people would see as being the size of baseballs, resulted in widespread crop damage during what should be the growing season. Then, of course, it skipped fall (which the summer was in the US) and went straight into winter.
That winter was especially harsh. According to reports from both Europe and the newly independent United States, it was one that not even this record 2014-15 winter, which is clearly one for the record books thus far, could even come close to topping. Temperatures were, like they are now, below zero in the eastern US for a good chunk of the winter. In fact, temperatures got so unspeakably cold that even the Gulf of Mexico froze! Crops failed as late as May, and in Europe, famine was widespread. Across the globe, it's estimated that more than 6 million people perished from the climatic impact of this seemingly gentle eruption.
Meanwhile, in the tropics, the Laki SO2 had other, far different (albeit familiar) effects. Thanks to the unusually extreme European cold, dry air began sweeping over Africa, over the Sahara, cutting off the African Easterly Jet and weakening the trade winds. Over Asia, that cold, dirty air made the high pressure over Tibet persist, causing the Indian Monsoon to also fail. In the Pacific, however, the effect was the opposite. Because the Atlantic was cold and so was the East Indian Ocean, the Pacific warmed up like wildfire. A super-strong multi-year El Niño began a few years after the eruption (presumably because there were a whole lot of factors that had to change prior to the El Niño, unlike in the case of today where we've got an El Niño that's already three years overdue), in 1789, and didn't end until 1793, causing flooding in California (then a Spanish colony; I would love to see if I could find some Spanish records of the storms, especially given that I learned almost 3 years of Spanish as a second language in high school) and South America that would be for the record books, for sure.
See, sulfur dioxide and hydrogen fluoride are acid gases. The mechanism of their toxicity is far different from the mechanism of action of gases like carbon monoxide that just put you to sleep for good. No, these gases will simply burn your lungs into oblivion if you try to breathe them. That's because when they contact water (which our lungs are full of), they turn that water into, respectively, sulfuric acid and hydrofluoric acid. Those strong, highly corrosive acids, then, are what do most of the damage. They can cause everything from wheezing and coughing all the way to pulmonary edema, which makes it excruciatingly painful to breathe, leading to death. That same property ― turning water into acid ― however, is also what makes these gases hard on the climate, potentially even more so than the ash. Why? When that acid happens to be acid high up in the atmosphere ― that is, in the clouds ― it's going to take a whole lot more than just a little sunlight to remove it. The reason? Its high albedo. Compared to water droplets, acid droplets are far more reflective and can persist far longer at the upper levels. They can take on rather bizarre yellowish or greenish colors, acting like a smokescreen that can drift to very high, even stratospheric levels. This has the effect of reflecting sunlight that would otherwise reach the surface and warm us up right back out into space.
Of course, that albedo-increasing property bring us right back to Bardarbunga. Although the eruption at Holuhraun was only 6 months long compared to 8 months for Laki, it's still probably at least one of several reasons why this current winter simply doesn't want to end. That is due, of course, to the extreme quantities of sulfur dioxide that it managed to release into the atmosphere, which was estimated to have peaked at a whopping 35,000 metric tons per day for 184 days, totaling 6.44 million metric tons of SO2. People have smelled Bardarbunga sulfur as far away as Norway, and SO2 concentrations in Austria have been in some cases higher than they were in the 1980's as a result of this eruption, believed to be emitting twice the amount of sulfur dioxide of all of Europe's power plant and oil refinery smokestacks combined.
That's a tremendous amount of reflective SO2 aerosols, if you ask me. And, of course, with now sulfur-coated Europe being so close to North Africa, the African Easterly Jet is now also likely to be affected. Yeah, that in turn allows for strong wind shear in the Atlantic, and from there, well, I think we all know what that means. Less hurricanes in the Atlantic but more in the eastern Pacific, which is of course abnormally warm... and if the effect on the Indian Monsoon is also one inducing a failure, it's likely that the resulting SST imbalance between the Indian and Pacific Oceans should also be enough to cause westerly wind bursts to ramp up like wildfire, which of course they're already trying to, and there's also already a Kelvin wave in progress as icing on the cake. Then again, we'll never really know until we actually see the activity...