I've only seen this done from a local database with commercial software, possibly because of the licensing restrictions the Royal Mail put on their postcode data.
There are APIs with reasonably high rate limits that can geocode UK postcodes, though. Yahoo's geocoding API is restricted to 5,000 hits a day, but that means you could have your entire database done in four days, store the lat/lon in the database, and just use it to add new addresses as they come in, say.
Also -- how accurate do you need the information? If you just need a rough location, you could geocode to the Outbound (e.g. "BS1"); I've seen user-sourced databases of at least arguable legality around the net for that.
