A lot depends on where you are travelling to, and returning from, the age of the child(ren) and the person taking them. Families are much more varied today than they have been traditionally, and many families have differing names, travelling schedules and members living abroad.
The best advice that I can give is to contact the embassy of the country you are travelling to or through, the airline or the FCDO for the latest advice. All I can tell you for sure is what I have done in the past, and it is rare for me to receive any feedback on whether or not the consent letter was looked at, or even whether or not it had to be notarised. Sometimes we need to get these letters apostilled which adds time and cost to the procedure.
In my experience Disney cruises need a parent’s consent when neither parent is travelling. I have had requests for set forms in Romanian to be completed when travelling to and from there. Teenagers have sought parental consent when going on holiday with their friends for the first time.
Recently I have seen an increase in notarised consent forms for school trips. This is typically where the child doesn’t hold a UK passport and the school are travelling to Spain for example. Notarised consent letters add time and cost to a visa application process, so please allow yourself enough time to ask the questions and obtain the answers.
If you would like to discuss your situation further, please don’t hesitate to contact me.