TL;DR - Without Britain, Germany wins both World Wars and becomes the new world superpower, the British Empire slowly falls apart, and large-scale advanced radar development gets set back a few decadesyears.
I disagree with Scott as far as his response regarding what the world would perceive the cause to be. There's no reason why the world would suspect it to be anything other than what it was - a natural catastrophe. Part of the reason nobody knew much about what caused the Tunguska event (at least for a short while) was that it was in rural Russia - according to Wikipedia, it took 13 years for the first expedition to reach the site. Given the importance of the target - after all, London was the seat of one of the largest empires the world has ever seen - I would think there would be a comprehensive investigation within a few weeks. But you can never properly tell when it comes to bureaucracy. Not that there would be much of that left in London. . .
World War I
World War II
So no Britain could mean the Triple Alliance wins the war. The big implication here is that there will be no Treaty of Versailles - at least not the same type of treaty that we are familiar with. This means that the seeds of anger in Germany that led in part to World War II would not be sown. Does this mean that there wouldn't be a World War II? Certainly not. Their ideologies would still develop. The Nazis would still have a fighting shot at taking power (although it would be difficult because many Germans wouldn't be angry at the result of World War I), and even though they might not be able to rally as much public support, I have no doubt the Holocaust would still have taken place. Czechoslovakia, Austria, and Poland all fall to Germany. There's no Neville Chamberlain to take the blame for letting things get out of hand, but they would still get out of hand.
It should be noted that Germany might not have gone through the infamous hyperinflation episode of the early 1920s, which has been attributed partly to the reparation payments in the Treaty of Versailles. Even if they had suffered through it, their economy would be stronger and would possibly have better taken the brunt of it, thus not laving Germany floundering economically.
Post-World War II
What about postwar? Well, Germany won't lose any of its top scientists to Russia, so the Soviet Union might not be as strong as they were. The US would also have a negligible rocket program (Werner von Braun would not be on the American side of the Atlantic) and Germany would have a decent nuclear weapons program. They could be the dominant power in the world.
Technology
As to what technology would not develop. . . Computers would be fine. ENIAC would probably exist (if the US entered the war), as would its German counterparts. Rockets would develop, although they would be German. I don't think a lack of Britain would hinder much other technological development, except for radar, which good old Hugh Dowding pushed so much for. Sure, it would have developed, but Robert Watson-Watt might never have lived.
As Mark pointed out, Britain was clearly not the only nation that was developing radar. However, Watson-Watt's work laid the groundwork for large-scale radar development. The Chain Home system was implemented in the mid-1930s, and proved to be crucial to the RAF's success in the Battle of Britain. This pioneered the usage of large-scale radar structures and advanced techniques.
British Empire
Finally, we get to the British Empire. It would not instantaneously collapse. After all, there were many British troops across the world, and many other people (primarily Britons or descendants of Britons) loyal to the Crown. Perhaps in a few months India would gain independence, as would many African nations. Australia, already on the verge of independence, would have become fully independent and become the country - continent - that we know today. But history would have changed a lot for pretty much every other British overseas territory. Oh, and Ireland would have had a lot more than Home Rule.
Does anyone remember the folks back in Scotland? I thought not. I think we all made the assumption that Britain would be completely floundering after losing essentially all of southern England, and we would be right. But Scotland (and Northern England, and Ireland, and Wales) would be relatively intact. Who takes power? In Scotland in 1908, there was still heavy influence from London, and the government was really just the overarching government of Great Britain. It is uncertain as to who would step up. Perhaps a semi-Cormwell-esque political figure would arise to try to take control and steer Scotland into a new non-monarchial direction. Scotland could also take control over all of Britain - although Wales would have a shot, too. I would guess the Irish would rather just live and let live, and stay out of it all.
