I can fully believe that staff members might be trained to not ask for Christian names, yes. However, I'm the least woke person on the planet but I probably wouldn't use the word Christian in that context either. I'd have said 'first name,' not because it's more politically correct and less potentially offensive to people of other religions but because it's just more up to date. We can't assume anyone is a Christian, regardless of whether they appear to be white, or sound British. Most British people probably describe themselves as atheists these days.
To be clear, I don't reject the idea of using the phrase 'christian name' but I think saying 'first name' has been standard practice and equally acceptable and interchangeable for probably two or three generations. Fifteen years ago I'm not sure anyone would have felt remotely motivated to be annoyed by that, not even a devout Christian.
But since, in recent years, we've been having our speech and our social mores constantly policed for 'microaggressions', or signs of 'white privilege' or 'colonialist attitudes' or 'unintentional bias' or plain old racism, it's not surprising that things like that have become a bit of a trigger for some people to feel affronted. It's just one more signifier of the erosion of our own culture to make space for other people's.