Many sites, including the new Gov.UK, use a font called Gill Sans. It’s a pretty font.
Problem is, like many good fonts, it’s not free. However, the creative right to the design expired in 2010 and are now in the public domain, so what isn’t free are the particular digital representations of the font.
I found a free one called Sans Guilt which may work well (I haven’t tried it out yet).
Also, some other blog I read suggested that Molengo, which is available from Google Fonts, is a very close representation of Gill Sans.
I may update this blog when I have tried them out and discovered how well they actually work.