It's definitely British. Dictionary.com gives the second definition as British; a washcloth.
Yet if you look at the standard definition on this page it is more particularly a type of fabric.
https://www.dictionary.com/browse/flannel
noun
a soft, slightly napped fabric of wool or wool and another fiber, used for trousers, jackets, shirts, etc.
a soft, warm, light fabric of cotton or cotton and another fiber, thickly napped on one side and used for sleepwear, undergarments, sheets, etc.
flannels,
an outer garment, especially trousers, made of flannel.
woolen undergarments.