Absolutely depends on where you are going. Most of the operators offer a package that will include UK. Three have a like home package is great if you are going to UK, Italy, Austria, Denmark and Sweden which allows you use your credit in those countries (if attached to Three network). Vodafone is more widespread network wise but it is 2€ per day for 50Mb. O2 also offer bundles like 50€ per month for 500Mb. Its all a bit stingy and certainly don't even think you can stream. I really don't know of anything that you could call Europe wide easy mobile broadband roaming available currently.
If you can find wireless hotspots you'll find it easier and cheaper. Otherwise get a local sim in the countries you are visiting and use it in an unlocked dongle.