As I have said MySudo is a good option. It is not well advertised but they do support people living in the UK and they provide UK numbers, although, you cannot get virtual cards as these are exclusive to the US.
You can also go into an O2 store and get a pre-paid sim. When you want to top-up you can go in-store and buy a voucher with cash which you can use to top-up your sim without any personal information.