A waistcoat (prounced more like "weskit") was a sleeveless vest as you see in the movies etc. Entirely taking off your tailcoat (worn over the waistcoat) was somewhat casual, probably a little too informal for most occasions on which Marianne would be likely to meet the Col.; but leaving your tailcoat unbuttoned, and so exposing the front of your waistcoat beneath, was quite common...

Flannel is an open woolen textile, usually without a nap. Flannels in JA's time would mean underlothing, bandages, or garments made of flannel. Not a very smart material, I should think. (Byron wore white quilted waistcoats, which he bought in immense quantity.)

In Victorian times, certainly (and possibly earlier?) flannel was worn next to the skin because it was supposed to confer special protective qualities against chills. Red flannel was thought to be especially effective, on some sort of colour therapy principle, no doubt.

However, this is not to be confused with hot flannel, a drink of gin and beer, heated after the addition of sugar and nutmeg (current in JA's time). Nor with the colloquial habit of flanneling so as to obscure one's meaning (don't know if this was current in JA's time).