Would there be a POD that would allow for a CDU equivalent to emerge in the UK?
There's a few major hindrances to this.
1) Christian Democracy tends to spring from societies where the Church is intrinsically linked with social provision, whereas parish councils and the like had been secularised in Britain a long time before Christian Democracy really became a thing. The most fertile societies are 'pillarised' ones, where you can spend your entire life in close contact with the Church - Christian trade unions, Christian schools, Christian universities, Christian sports clubs (in the Netherlands, Catholics played football on Sundays and Protestants played on Saturdays) and, as an outgrowth of these things, Christian political parties. This did not exist in the UK in the early 20th century (note also that France is similarly bereft of a CDU equivalent, because they introduced laicite previous to Rerum Novarum).
2) The social market economy and solidarism are the core economic tenets of Christian Democracy, and are seen as being a third pole, distinct from both socialism and laissez-faire liberalism. In the UK, we tend not to see it that way, and all alternative economic theories are judged to be somewhere on the scale between Right and Left. See the reaction to Liz Kendall's proposal of a social market economy in 2015, which was largely met with "why are you a tory now". Changing this probably requires PR, but it might also be integral to the Anglo psyche.
3) Christian Democracy tends to come from the Catholic side of the aisle, which has historically been a lot more involved in the daily lives of its members than Protestantism (see point 1). So in Germany, Zentrum and the CDU have traditionally been the party of the Catholic middle class, while most other countries with strong Christian Democratic parties tend to be those which are predominantly Catholic. There are, naturally, exceptions - Switzerland (but its Christian Democrats are mostly Catholic, with the exception of the small Evangelical People's Party), the Scandinavian countries (but these are very small parties whose christianity is based largely on Culture War issues) and the Netherlands, which is just weird. Catholicism is obviously quite rare in the UK, so you're looking at this CDU being strongest in Glasgow, Merseyside and Northern Ireland - side note: I wonder if the NI Nationalist Party could be characterised as Christian Democratic? There's certainly a case for certain sections of the SDLP.
So without solving these problems, you're probably not going to get a CDU even if the Tories shit the bed at the appropriate time. What you might get is what I would characterise as Protestant Christian Democracy (e.g. the Norwegian and Swedish Christian Democrats, the small Dutch parties CU and SGP, the Swiss EVP, the NZ and Canadian Christian Heritage Parties, the Australian Family First and others, and the American Evangelical Right - all but the last of which, you will notice, are very small and only survive due to PR where they survive at all). Protestant Christian Democracy co-opts Christian Democrat economic policy, a cheeky bit of environmentalism - ahem, 'stewardship' - from the Greens, and a whole heap of sex-related hatred, which is their main focus. This is what we have IOTL with the Christian Peoples Party, the Christian Party, David Alton and various others, but obviously without any major success.