If you haven't come across this yet, then there is still time to sign up.
GOV.UK are opening up the tendering and bidding process for smaller companies in the name of transparency and reducing costs. Traditionally it was larger companies that tended to hoover up all the government contracts, but now anyone who can provide a relevant service can compete.
This is aimed at companies providing digital services, but not just web designers, it can be project management, infrastructure, design, training etc.
If you are interested, then follow the below link and follow the sign up process. If your company meets the relevant condition's, then you maybe listed on the governments procurement list.
We did a similar thing with the Scottish Government, and it does work and you can win contracts.
Its open to all business types including sole traders, so what have you got to lose?