I had never thought of it as being a uniquely British thing but then compared the top 100 names for the US and the UK and you're right... only in the UK do you see
Millie,
Ellie,
Lexi,
Gracie,
Libby,
Tilly, etc.
It's usually younger mums, but not always.
I suppose itremoves the annoyance of having to correct teachers, etc when they call you bby your full name but you never ever go by that.
But personally I would like my child to have a more formal-sounding name to fall back on when they need it.
S xx