On my own booking, I made my changes at the beginning of September. It wasnt reissued until 4 months later, and that's with me calling regularly to make sure my credit card security code was stored (it drops out after 7 days).
Since you're in the UK they'll probably call you if the need the CVV again, then the reissue will likely be done there and then. I'm in the US and they only ever sent an email asking me to call them...