For what it's worth, Hanover wasn't a surname. It was the name of the state the kings came from. In Germany, they used the family name (Welf or Guelph; both versions were used). Same goes for their Saxe-Coburg successors (family name Wettin). The change in name was prompted by an article by H. G. Wells in which he said something about "this alien and uninspiring monarchy". George V growled that "I might be uninspiring but I'm not alien" and promptly changed the name of his house to that of his favorite castle. A different king could easily pick a different name.
It's more likely than not that Britain wouldn't get sucked into the war between Austria and Prussia in 1866: Britain had favored Denmark in the Schleswig-Holstein conflict in 1864, and had found itself unable to provide any meaningful assistance, so why get humiliated again. In that case, the king would remain king of Hanover, too, but Prussia would dominate Germany as in OTL. If Britain did try to intervene on the Austrian side in 1866 or the French in 1870, it's unlikely to change the outcome, so dominant were the Prussians in both conflicts. Eventually there would be a peace treaty; possibly the king of Britain would retain retain all or part of Hanover or Britain would be compensated by territories outside Europe.