What political groups were around in the time of Oliver Twist?
The mid-nineteenth century in England was shaped by the Napoleonic wars, which ended in 1815. [Dickens' Oliver Twist was published in 1938; Queen Victoria took the crown in 1937; the Poor Laws were passed in 1834.] The wars, England's fight against France, resulted in a heavy burden of taxation on the poor classes leading to economic instability that itself led to wide unemployment. Inflation resulted and was coupled with a closed market for English exports to European countries.
In conjunction with this, industrialization displaced workers and compounded unemployment due to the replacing of human labor with machine labor. Things became so bad, that being a Luddite--angry worker leading movements to crush machinery--became a crime punishable by death in 1811. On top of this, children were employed in factories to work the machinery because they could be given a lower wage and because their small sizes gave them access to intricate machine parts. Parents yielded because adults were themselves out of work.
In this milieu the political groups in England were the dominant Whigs and Torries, but in mid-nineteenth century England, they could not yet be called parties in the sense recognized today as they were more in the nature of alliances of like-minded individuals. The Whigs were then associated with the newly risen industrialization middle classes with new money whereas the Torries were then associated with landed gentry and the Churches of England and Scotland.