I wasn't sure even whether to post this as I don't know the details as I'm not there, by my dad got the vaccine on Tuesday or Wednesday. He's only 72, so technically not in the target group yet, but according to him, the south-east of London, in this case Sidcup*, has spare slots and is filling them. He called Guys and got a slot the next day, but they wouldn't do my mum, who is 68.
I don't want this to be gossip-mongering or spreading fake news, I'm just going by what they've said, but it might be helpful.
* It fucking is London, alright.