No, it wasn't because it was a child saying it. It's because in (British) English the 'correct' way to greet someone you have never met before is to say 'How do you do', not 'Pleased to meet you'.
The Daily Mirror has a rather tongue-in-cheek article about how to tell whether you're 'Posh', and using the phrase 'Pleased to meet you' is one of the key indicators that you're not. (Actually, I think reading the Daily Mirror means that you're automatically NOT Posh, but that's just my opinion.)
The terms 'U' (upper class) and 'non-U' (not upper class) were used to differentiate the way the upper class spoke from how the middle class spoke when trying to be 'posh'. Wikipedia (not always a reliable source, I know), gives a list of U/non-U words and phrases. 'Pleased to meet you' is part of the 'non-U' vocabulary, and is (presumably) not something that an upper class person would/should say.
The Telegraph article was from 2010, and the author (Jessica Fellowes) was speaking about her childhood. She was born in 1974 (see Wikipedia), and evidently the U/non-U divide was still going on during her childhood. (She is the niece of Lord Fellowes, who wrote Downton Abbey. It's reasonable to assume that her mother would have wanted her daughter to speak in a way that showed she was a well-brought-up young girl.)
This sort of etiquette is old fashioned these days, and is only likely to be of much use if you are mingling with the upper class and attempting to pass yourself off as one of them (and in Britain, unless you've studied the rules very carefully, you're likely to give yourself away in many different ways before you're even introduced to someone). In general polite conversation, it's absolutely fine to say 'Pleased to meet you'.