At one point in time, the Parliament was dominated by the nobility but nowadays it's dominated by the "common man" (though a cynic may say it hasn't changed at all).
My question, specifically, is basically when did it become OK for someone who wasn't born to noble blood to hold actual political power?
It was taken from the hands of the nobility in 1999 when Labour removed the rights of peers to sit in the House of Lords, which is the UK's second chamber.
92 hereditary peers are left there today, along with 26 Princes of the Church. The rest are members of political parties that were ushered while their party was in power. Sixteen years later it's still a controversial issue and reeks of corruption to many.
FYI, this subreddit has a rule that excludes everything that happened under twenty years ago.
hi! this FAQ section may be more narrow than what you're looking for, but you may find something useful here
There were a series of changes over the nineteenth century and into the early twentieth.
There was a shift in economic and social power towards the middle class which had gained from the industrial revolution. This led to demands for representation. There were a series of electoral reforms from 1832 to 1918 which expanded the electorate from about 3% of the population to something close to full suffrage. Constituencies were redrawn to reflect population so cities became more important. The Secret Ballot was introduced in 1872 which made it harder for landlords to control their tenants.
Despite this, nobles were important into the late 19th century. Lord Salisbury was Prime Minister until 1902 and it was normal for Lords to serve in Cabinet.
Other aspects of British society became more egalitarian with exams for civil service and military positions reducing the role of patronage.
In 1909 the House of Lords vetoed the Liberal government's budget leading to a crisis At the end of this, the Lords was left with no veto and could only delay bills.
At the same time, the landed aristocracy were in economic and social decline. This was exacerbated by WWI and the economic crisis which followed.
By 1963 when Lord Home succeeded Macmillan as PM, he resigned his peerage and served as Alec Home as it was seen as wrong to have a peer as PM.
It is hard to summarise this without falling into the Whig model of the inevitable rise of British democracy, it was actually a more complex process with a lot of expedient decisions which had unexpected outcomes.
Sources
David Cannadine, The Decline and Fall of the British Aristocracy
It was something of a gradual process, as the Commons asserted progressively more power after the Commonwealth (1649) and the Glorious Revolution (1689). However, even then the government tended to be led by members of the aristocracy and armed forces. Of course, it depends very much on what you mean by common man!
The first actual commoner I can think of is Pitt the Elder in 1734, although he was from an extremely wealthy family that had made its fortune in India, and he was elected to the 'rotten borough' of Old Sarum when his brother passed it to him. Old Sarum was basically a hill near Salisbury that, at the time of its abolition, returned 2 MPs for a population of 11 voters, all of whom were landowners elsewhere. Pitt was known for a long time as The Great Commoner due to his refusal to take a title, but eventually relented and was created Earl Chatham.
The picture becomes slightly more clear after the Act of Union 1801, which merged the parliaments of Great Britain and Ireland. In practice, this still restricted who could sit, as the Oath of Supremacy required members to swear an oath of allegiance to the monarch as head of the church. (Similarly, holders of governmental positions and students at university were required to be members of the Anglican church). This restricted parliamentary power to Anglicans, which was particularly controversial in Ireland because they were a minority. Eventually, a series of campaigns resulted in the repeal of the Test Acts and Catholic emancipation in 1828-9, opening government positions to catholics.
One of the first beneficiaries was Daniel O'Connell, a catholic lawyer who had been elected in 1828 but unable to take his seat until after emancipation. O'Connell had originally been of noble stock, but his family had been dispossed of their property by the Penal Laws over a century before. In the eyes of the English establishment, he would certainly have been a commoner, so he could be a candidate as well.
However, there's still the problem of those rotten boroughs like Old Sarum. The electoral system still put power in the hands of wealthy landowners, who could choose their own candidates and essentially buy seats in parliament. The Great Reform Act of 1832 introduced a wide-ranging series of changes to elections; the rotten boroughs were disenfranchised, constituencies were redrawn to reflect the demographic changes (large cities like Manchester and Birmingham had no representation compared with Old Sarum, which elected 2 MPs!) with single members returned for each constituency, and slightly extended the franchise. This brought a raft of solidly middle class members to power, including Disraeli and Gladstone.
As the nineteenth century progressed, there was a large increase in the number of working class members. West Ham South is one constituency that leaps out - it returned Joseph Leicester, a glass blower, in 1850, and the Scottish Labour politician Keir Hardie in 1892. Hardie is perhaps the best early example of the 'common man' I presume you're talking about - born to a poor family in Scotland's central belt, a former miner, trade unionist, and journal ist, he turned up to parliament (to much derision) in his plain tweed suit and deerstalker hat.
After that, you're looking at perhaps 1928, when the franchise was extended equally to women, when the first common members, democratically elected by commoners, were returned to parliament.