Honestly, a Victorian Era British Army with continental commitments is a massive about turn IMO, so much so that in order to prevent causing severe economic problems (labour shortages primarily, as well as fiscal costs) the British will have to turn to India and the colonies. Which changes everything.
I agree with the idea of introducing better guns earlier, but needing to effectively move from "Rule" to "Play Nice" with the subcontinent demands significant changes to the Empire, as much from the need for large, loyal, British-Indian regiments that are less costly, to the demands of India as a result.
Lets just say that we have the British resort to forming new Indian Forces for deployment. Usefully, they aren't going to side with Germans in Germany - unhelpfully reinforcements are quite far away.
However, whilst forming new Indian forces that can be deployed in Germany is cheaper than British forces, this still impacts India - who'll have more influence via the dependence on their people for troops. More self-government, etc. Which could undermine the British economic model without a shift to effectively treating India as a Tax Colony rather than a Resource Colony.
A weird aside is that this would likely reduce the impact of future famines, because not only is there more self-rule, so better policy and the ability to prevent money-famines, but there is also the impact of soldiers sending money home to reduce money-famines as well - and the political impact of having soldiers families starving should be obvious.
So yeah, the biggest impact of Hanover being a British commitment IMO would be the large-scale transformation required to effectively supply troops in numbers capable of intimidating the French and Prussians, a shift to a form of Federal Empire around Tax Colonies rather than OTLs Empire.