In the UK everyone I know uses the abbreviation bra for this garment. If (rarely) they feel that they have to use the full word, they attempt the French pronunciation: like a 'brassy [as in the North of England] air' IPA: /ˈbræ zi er/. In the United States, the abbreviation is less used, and the attempt to sound French is often further from the original: 'bre-ZEER' IPA: /brə ˈziːr/.
Oddly enough, the French word is soutien-gorge, not 'brassiere' - not even with the accent brassière. In French, the word brassière means a 'shoulder strap', as for example in a back-pack; 'a bodice [particularly for children]'; or '[a toddler's] reins, or leading strings'.