I'm not sure which time you're referring to here, although I think the basic answer is the same either way. Do you mean the era of the French Revolution, when the Bourbon dynasty was deposed in France, followed by the collapse of monarchies across the Continent as a result of Napoleon's conquests? Or do you mean the era after World War I, when the Central and East European monarchical regimes came down?

In Either case, the answer is largely the same. First, the British were on the winning side in both those conflicts, meaning that no one forced the royals to step down, a la the German or Hapsburg emperors. It also helped that the monarchs during both of these episodes were relatively ineffectual, with George III largely incapacitated and George V comparatively disinterested in politics and willing to let the parliamentary government handle things. That's not to say that there weren't threats. While the threat of revolution in Britain was never very serious, there were republican radicals in the early 19th century and a strong 'Independent Labor Party' and some Communist agitation after World War I. But especially in the latter period, the monarchy had become a more popular institution (meaning it played a mostly symbolic role) and wielded little real power, especially after the reforms to the House of Lords eliminated the hereditary element of society's ability to veto social and political reforms.

The other explanation, however, is that the British monarchy didn't survive. It actually went down earlier than any of those I listed above, but for that reason was able to make a comeback. Afterall, the British monarchy hadn't been really British since the 1600s. The English Civil War ended in 1649 with the establishment of a republic and the temporary overthrow of the Stuart dynasty. Then, when Charles II Stuart is brought back in 1660, he dies without an heir and his brother James II takes the throne. James II is believed to be Catholic (a problem in Reformation Britain) and is certainly absolutist (a problem given the robust parliamentary system) and so he is chased from power in 1688 in the 'Glorious Revolution.' William, who was Dutch, and Mary, who was Charles II's niece, rule Britain after this and were succeeded by Mary's sister Anne, who was really the last fully English monarch. After this come the Hannovarians who, as the name denotes, came from Hannover in northern Germany. There continue to be struggles between the monarchs and parliament, but they are increasingly won by the latter, as the Lords' Reform mentioned above indicates.

So, you could argue that the monarchy, as an institution, survives partly because it was lucky enough to be on the winning side, partly because it was greatly weakened by the 17th century revolutions that placed Parliament more firmly in control than anywhere else in Europe. Or you could argue that the British monarchy underwent a turbulent disestablishment just like everywhere else, but managed to hold onto an increasingly ceremonial role, simply by learning to keep their heads down.