"And in this age of disposible Pampers and Huggy diapers, are 'nappies' still used in the UK?"

The word nappy is still used for either the Huggies-type disposable or less used Terry-towelling type in the way you use the word "diaper".
As for "Natty", whilst not in common use, it is a familiar, if old fashioned word.
"That's a natty tie you're wearing" would usually be meant as a compliment to your good taste, although with older/unfashionable words or phrases this can often be turned on its head and be used sarcastically as a criticism.
An example I've come across recently is young people using the word "groovy" as a mocking description of something not trendy or of bad taste.
"Hmmm groovy shoes!! HAHA"