I wouldn't characterize post-Magna Carta England as having a weak central government. Compared to the Holy Roman Empire it had a very efficient central government, in which the parliament played an important role alongeside the king.
The early English Parliament already had a House of Commons. Hence not only the nobility was given rights but the common people too. Of course, the elections to the House of Commons were far from democractic in the modern sense, with voting rights limited by land or property ownership requirements.
Besides the Magna Carta in 1215, a number of key events in the development of English/British democracy must be noted.
The Second Baron's War in 1264 led to the creation of Simon de Monfort's "model parliament". Although the rebellion eventually failed, the parliament set a precedent for the future.
The English Civil War in 1642 led to the temporary establishment of the Commonwealth of England, a republic. Besides early republican ideals, an important factor in the revolution was religious (Puritan). The Restoration in 1660 returned the monarchy but the power balance permanently shifted towards the parliament.
The Glorious Revolution of 1688 resulted in the establishment of William III and Mary II in place of James II. The further (significant) weakening of the monarchy was codified in the Bill of Rights of 1689. The Glorious Revolution marked a clear victory of the social contract theory of government over the divine right of kings. The formation of proto-democratic ideology accelerated due to the efforts of thinkers like John Locke.
The Reform Act 1832 widened electoral suffrage significantly and eliminated corrupted practices such as "rotten boroughs" and "pocket boroughs" which allowed a small group of powerful men to control a large number of seats in the Parliament. This was already after the American and French revolutions, and republican ideals were becoming widespread in the western world. England has had a large number of its own liberal thinkers such as the radical Richard Price and the more conservative Edmund Burke. Further electoral reform was stalled for a while although promoted by movements such as Chartism (founded by the People's Charter of 1838). Eventually reform continued with the Reform Acts of 1867 and 1884.
Representation of the People Act 1918 granted almost universal male suffrage and limited female suffrage.
Representation of the People Act 1928 granted females and males equal voting rights.