You might do best to identify a few issues you'd like songs about, and search sources like the Bodleian Library ballad database for appropriate items.

There are an absolute shitload of songs about the Reform Bill of 1832 and the elections that immediately followed it. I know the Edinburgh ones (there are perhaps 100, some of them astonishingly obscene), but every town in Britain will have had them. There are a few songs about the cholera epidemic of the same few years (less than you'd think). Stacks of songs about the Crimean War, mostly satires on government bumbling. A lot of broadsides about the radical agitation of the years 1815-1820. Chartist songs. Internationalist political songs about the events of 1848. Dickens himself, as a journalist, covered a dinner in honour of Earl Grey, the architect of the 1832 electoral reform; there is a strathspey by James Hill commemorating the erection of a monument to Grey in Newcastle about 20 years later.

There must be loads of topical ballads about the Great Stink of 1858, look at the newspapers of the time - most of them had a topical song or poem in every issue.