What changes do you see then?
First of all the heir at the time would have been Ernest Augustus, he didn't die until 1841. So a brief rule of Ernest Augustus I of Great Britain and Ireland followed by his son George V. Both men were conservative autocrats, it was Ernest Augustus who ripped up the liberal constitution created for Hanover by his brother the Duke of Cambridge during his stint as Viceroy.
Now Great Britain as a nation is a much bigger prize than Hannover, and George V wouldn't accept Hanover just being outright invaded by a rising Prussia, however the monarchy of Britain had largely lost it's overt political power and the British Parliament was never happy at having to deal with Hanover. Having seemingly got out from underneath Hanover with Victoria and now having to deal with it again under a new string of Kings would be VERY annoying for them.
However just letting Prussia walk in and take it without challenge would look bad for Britain, especially since it has a sprawling colonial empire held together by fear of British force, any sign of weakness could see a domino effect of colonial rebellions all around the globe as news spreads.
At the time however Britain's actual army was bad shape, the later Crimean War would expose just how outdated, poorly equipped, and trained the actual British army is. Less developed societies in Africa and Asia are something they can handle, their fellow Europeans... Not so much.
The Crimean War was fought from 1853 to 1856 and the aftermath saw Great Britain slowly begin reforming it's military (though it wouldn't be straightforward). The Austro-Prussian War that saw (among other things) Hanover become part of the emerging German Empire was in 1866.
I could see Britain begrudgingly joining in the conflict, but it's performance would be messy and further expose that the British military is still in need of reform and in need of Professionalism. Hanover would still wind up annexed and Britain and it's Hanover King humiliated.