TLDR: Yes, but not as common as remembered now.

The actual matter of the signs themselves is attested to in the work of Britain’s race relations scholars, who were a group of sociologists and anthropologists keenly interested in how diversity was affecting Britain. They tended to parachute into immigrant neighborhoods in London (e.g., Stepney, Brixton), Birmingham, and Manchester; there’s an interesting literature on the way that this scholarship, which was meant to be sensitive to the struggles that migrants faced, ultimately reinforced stereotypes of migrants, especially Caribbean migrants, as “dark strangers” who couldn’t adapt to English life.[1] But for our purposes, they do note the presence of anti-Irish signs and discrimination in housing, as this sociologist noted:

In some districts a fairly active discrimination over housing is enforced against the Irish. In Birmingham, Manchester and parts of London it is not uncommon to find notices offering accommodation that specify ‘no coloured; no Irish’. There is a strongly held impression that these groups, once they move into a street or district, will ‘lower the tone of the neighbourhood’ and, what is much more important, will ‘bring down property values.’ [2]

Of course, this brings us back to your original question: were the signs common? This is one also asked by Irish historian Enda Delaney, in the third chapter of The Irish in Postwar Britain, and he concludes:

“No blacks, no Irish’ or ‘No Irish need apply’ signs were apparently not uncommon sights in the 1950s. In fact, they occupy a central place in the collective memory of the Irish in Britain, often featuring in the personal testimony of migrants as emblematic of the reaction to large‐scale Irish settlement in post‐war Britain and even appearing in the title of the autobiography of John Lydon, the lead singer with the punk band, the Sex Pistols, who was raised by Irish parents. Such signs are, by their very nature, ephemeral, but unlike the American case, where the existence of ‘No Irish need apply’ signs was challenged by one historian, who claimed that Irish Americans had incorporated this myth into a wider account of discrimination and anti‐Irish prejudice, no one has, as yet, set themselves the almost impossible task of disproving their existence. But caution is needed here as such foundational narratives can effortlessly enter oral testimonies as individuals interweave a well‐known communal memory into a personal life story. There is little doubt that the terminology was subsequently altered to take account of more recent nomenclature—‘coloured’ was the term in widespread use before this description came to be seen as offensive in the 1970s.” [3]

So were there signs? Sure. But it’s more likely that the signs have come to stand in for the more insidious forms of discrimination in housing, and in Britain writ large: long waiting lists for council housing, which put all immigrants on waiting lists based on residency; or council housing that was poorly suited for the needs of applicants; or private landlords and ladies who turned away Irish applicants because “the neighbors might object,” often based on stereotypes about what Irish migrants “were like” and “how they lived,” e.g., often in crowded lodging houses, which had nothing to do with “Irish culture” but most everything to do with the fact that they couldn’t afford or rent anywhere else for the time being.

The other point I would mention here is the ambivalent status of Irish migrants in Britain. Irish people have been migrating to Britain since the 19th century; British popular depictions of Irish migrants depicted them as poor, dirty intruders who undercut the true British working class and whose culture was different and inferior. So too in the 1950s. But the arrival of black migrants from the Commonwealth drew attention away from the Irish. As hostility towards black migrants increased, the Irish became deracialized and assimilated into the category of “white” in popular and official British understandings of migration. The grouping of "no blacks no Irish" became ungrouped, as Irish migrants were deemed acceptable (though not always equal), and black migrants, not.

You can begin to see this in one example provided by the sociologist Sheila Patterson, in her 1950s study of Brixton. She reported that landlords in Brixton described Irish migrants as “bad as the darkies”; but they are not so easily identifiable, and the criticisms are therefore not so sharply focused’.[4] Here, British landlords perceived Irish migrants, who were less visibly different, as more acceptable tenants than black migrants—the neighbors "couldn't object" to migrants they couldn't identify.

The distinction became written into immigration law, too. British policymakers offered Irish migrants free right of entry into the UK on the grounds that the Irish—unlike black migrants—were of the same race as Britons. As the historian Kathleen Paul writes, British policymakers regarded Irish migrants as "an 'unpredictable and inconsequent people,' according to a 1949 prime ministerial brief... [but] nevertheless regarded as worthy of continued access to Britain thanks to 'arguments of commonsense' and by virtue of the 'outstanding' fact of their race." Yet those same policymakers ensured that black migrants from the Commonwealth were subject to entry quotas just a few years later. [5]

By the next decade, the visibility of Irish migrants receded from public view even further, particularly as the very concepts of “migrant” and “minority” became associated with Caribbean, African, and South Asian migrants. In sectors with high #s of immigrant workers (e.g., health care), many administrators stopped counting Irish nurses as immigrants at all. The term “overseas nurse” became shorthand for nurses of color primarily—even though of course Irish nurses’ experience of work and training was shaped, often profoundly, by their Irishness.[6]

[1] On race relations scholars, see Chris Waters, “Dark Strangers in Our Midst: Discourses of Race and Nation in Britain, 1947-1963,” Journal of British Studies, 2007

[2] Quoted in Enda Delaney, The Irish in Postwar Britain (Oxford, 2007)

[3] Enda Delaney, The Irish in Postwar Britain, ch. 3

[4] Quoted in Delaney.

[5] Kathleen Paul, Whitewashing Britain: Race and Citizenship in the Postwar Era.

[6] Louise Ryan, “Who Do You Think You Are? Irish Nurses Encountering Ethnicity and Constructing Identity in Britain,” Ethnic and Racial Studies, 2007