Here we go with the anecdotes.
I travel 4-6 times a year with dd (different surname/Italian, it's usual here) and have been asked several times for letter of consent.
It is not specifically to do with different surnames, it's to do with one parent travelling alone with children who might be abduction those children cross-border.
For at least the last 30 years (since I worked in this field) border control are told to pay special attention to a single parent travelling with children.
You may be asked for a consent letter, you may not. Some countries (Canada and Mexico iirc) absolutely insist on it and it has to be signed and witnessed by a notary.
The children's b/certs are enough to establish parentage, they are not enough to establish consent. I imagine (and I asked an officer last summer when we came into Gatwick and he said he couldn't possibly comment but possibly) that sooner or later, there will be clearer guidelines about this, as at the moment checks are done on an ad-hoc basis.
Dd is now a teenager and on the last 4 entries into the UK has been asked where we are going. They ask her, not me. Up to her being about 10 they asked her if I was her mother. That's their preliminary check to asking for the docs.
Only twice have I been asked for the actual consent letter.
I think if you are no-contact, or the absent parent has no residential order thingy or is deceased, then a letter stating those things usually suffices if you are asked.
The irony of all this is that I take groups of Italian teenagers to the UK. Am I ever asked if I have permission to take them? Nope. Because they're all over 14. My own daughter? Yep. Because I may be using the school trip as cover for a cross border abduction. 