The Chartists didn't want to overthrow the monarchy, they wanted to reform Parliament. Also, a lot of the signatures on their petition turned out to be fraudulent so I don't think they were nearly as popular as claimed. For them to overthrow the Government would probably require something totally radical, such as a march turning ugly and ending up with the chance capture of Buckingham Palace and hot blood dictating actions. In such circumstances I think the Chartists would quickly lose their popular support and would be crushed by the military later. If somehow...somehow...their petition was granted (and even this is pushing it) then it would change British foreign policy, but it wouldn't change the reigning monarch and assuming a continuing UK-Hanoverian Personal Union I still think the Prussians would shy from attacking Hanover, even if the new Parliament was happy to consign Hanover to its fate.