In short?

The core of British thought has been, and always has since the wars of Louis XIV in the late 1600's and early 1700's, to not allow a single power or power bloc to dominate the continent. That is where Prussia comes in. From the war in the Netherlands with Louis XIV to the Seven Years' War to the Napoleonic Wars their cause was very clear: Do not let someone achieve continental dominance and to do that by backing those powers enemies.

Look at the Seven Years' War and the Napoleonic Wars; as late as the 1760's the British were outright opposing the Russians and Austrians (and French/Swedes) by supporting Prussia. Fast forward to 1803 and now the British literally pay Russia and Austria to go to war to contain the French. The joke "perfidious albion" would arise to describe their almost video game comical level of side switching -- they were allies of convenience to ensure Europe remained relatively fractured. Ultimately Britain and Prussia were not 'friends' they were allies of convenience. Britain looked to the seas and Prussia looked to the continent but their interests aligned -- beating France and her allies. For Britain this meant Hanover and mostly French colonial possessions and for Prussia this meant Silesia.

So why did Britain switch sides so to say again? Well because after the Napoleonic Wars Britain wanted to counteract Austrian influence on Germany and thus Prussia was given a sizable portion of West Germany. Fast forward another 50 years and Prussia just recently beat Austria in a war, browbeat Russia diplomatically, and beat France in a war as well and then united the entirety of Germany under her wing. As they made more moves under Willhelm II to achieve even more continental dominance Britain would support her main rival -- France. It was more of the same.