Most charity shops won't take used bras.
Bravissimo "bra banks"
The lingerie/swimwear chain Bravissimo runs a bra recycling scheme in shops. Every shop has a "bra bank" (usually in the changing room area) where you can leave your old bra. It doesn't matter what condition it's in! Those in a usable condition go to women in other countries, those that are unusable are recycled. Bravissimo also makes a 70p donation to Refuge for every kilo of donated bras. List of Bravissimo shops: https://www.bravissimo.com/shops/all/
Smalls for All: send bras by post
The charity Smalls for All takes unworn or "gently worn" bras and distributes them to people in Africa. Post them to:
Smalls for All 108 Buchanan Crescent Eliburn Livingston EH54 7EF
Source: http://www.smallsforall.org/get-involved/donate/, confirmed by email January 2018.