This might be more of a legal question than a historical one, but I'll give it a go. I'm working mostly off Andrew Blick's literature review literature review and the Parliamentary Political and Constitutional Reform Committee's last report.

Disclaimer: I'm not a laywer or a historian, I'm just very interested and in possession of a search engine and free time. Writing this has been very educational!

First, most existing codified constitutions are the results of revolutions or other political upheavals. See: the French and American constitutions, which were products of revolutions; the 23 different Haitian consitutions, the first of which was a result of the expulsion of the French; the various European constitutions, some of which were a result of the French invading and imposing variations on theirs, and some of which are results of more recent revolutions. It's difficult for a country to impose a codified constitution without a shake-up, because a codified constitution is generally a sign that something's gone horribly wrong with the previous version.

The Magna Carta can be considered the first attempt at an English consitutution (note: not British) as it placed formal limits on the monarch's power, and codified pre-existing legal tradition. It led, as is the pattern, to civil war.^(1) It establishes, first and foremost, that the King's will is subject to 'the will of the people,' i.e. that the King can't enact certain laws without the consent of Parliament, though at the time it was much harder for Parliament to disagree, and the provision mostly applied only to the levelling of taxes. It also did not apply in Scotland or Ireland, and initially did not apply in Wales or the Welsh Marches. ^(2)

The Magna Carta establishes the principle of parliamentary sovereignity. This means that the laws and acts of Parliament can be created and ended at any time; that Parliament can create any laws they feel like; and that no law can be created that cannot later be changed. Until 2009, no Parliamentary law could be changed by the courts, but the creation of the Supreme Court changed that principle (see how this works?). When people opposed the signing of the Human Rights Act, it wasn't because they disagreed with the provisions, but because it was a law that couldn't be edited by Parliment, only withdrawn from. (In practice, it's possible for Parliament to pass laws that violate the Act, but mostly they don't, for manners' sake.) Parliamentary sovereignity is one of the most fundamental parts of the British constitution.^(3,4) A codified constitution, if it were uneditable or even particularly difficult to edit, would breach this principle.

Another restriction on parliamentary sovereignity is when it comes to the other three countries. The UK parliament keeps some topics as 'reserved matters,' mostly international things like defence and immigration. Other than that, many things are legislated by the sub-parliaments of each devolved country. For instance, at the moment, each devolved nation is responsible for handling their own COVID-19 response. However, this, too, is applicable to parliamentary sovereignity; in theory, devolution can be rolled back. This will probably never be done, because that's how things like civil wars happen, again. Devolution has been a continuous part of British history; Wales and the March regions maintained different laws and parliaments in the fifteenth to seventeenth centuries, Northern England kept a Council for most of that same period, and until 1801 British-held Ireland had a legislative parliament, though it was very curtailed. Even London and other major cities maintain the right to certain executive privileges under the power of the mayor.^(5)

This is sort of like the federal system in the States, in which each state has a certain degree of autonomy, except that the Constitution applies to every state. A codified constitution would almost certainly have to overrule the authority of local and devolved authorities and countries in a similar way, which would be difficult to manage, to say the least. The PCRC's report, which I've linked above, has some sketched proposals for how to overcome this; it's up to you how satisfying you find them.

In short: a codified constitution would be seen, in legal terms, as a curtailment of democracy and local power, whereas the establishment of consitutions in other countries generally represents the opposite.

The last time that parliamentary sovereignity was abolished (rather than curtailed) was Cromwell's Commonwealth. The beginning of the Civil War began with a major restriction of royal power and the expansion of parliamentary power, but Cromwell's dictatorship asserted sovereignity over parliament. The reconvening of parliament and the return to traditional uncodified constitutional law came with another expansion of parliamentary sovereignity in reaction, and followed several experiments in alternate constitutions.^(6) This was prior to the French or American revolutions, and there wasn't a tradition of codified constitutions in democracies at the time. Perhaps if there were contemporary international examples, this would have gone differently, but that's speculation on my part.

A speech by Lord Hailsham, who had been Lord Chancellor from 1970-74, gave a speech in 1979 in which he said that it wouldn't be possible for a codified UK settlement to come about without 'a revolution, bloody or peacefully contrived.' He also suggests that the lack of a codified constitution protected Britain from the upheaval that would change it, as major constitutional changes, such as the outlawing of slavery, can be handled quickly (relatively) by Parliament, which a codified constitution might prevent.^(7)

There are very interesting arguments both for and against the codification of the constitution, a few of which I've mentioned, which I don't have the ability or the time to explain, and this is a historical question. Mostly, it appears that the reasons why we don't have a codified constitution (yet) are: we didn't undergo the sort of upheaval that results in one at a time when codification was in vogue, and; we didn't want one, due to certain legal principles.

[1] National Archives: http://www.nationalarchives.gov.uk/pathways/citizenship/citizen_subject/making_history_citizen.htm

[2] British Library, which houses one of the three existing first copies of the Magna Carta: https://www.webarchive.org.uk/wayback/archive/20131031165418/http://www.bl.uk/onlinegallery/takingliberties/staritems/21magnacarta.html

[3] Parliament: https://www.parliament.uk/about/how/role/sovereignty/

[4] British Institute of Human Rights: https://www.bihr.org.uk/thehumanrightsact

[5] House of Commons Research Briefing https://commonslibrary.parliament.uk/research-briefings/cbp-8599/

[6] National Archives again: http://www.nationalarchives.gov.uk/pathways/citizenship/rise_parliament/making_history_rise.htm

[7] BBC: https://genome.ch.bbc.co.uk/9474f2a73e814142ae93290b9e55bf72