I've injured my hand and is swollen. can make a loose fist but painful. could it be broken and is it worth going to a&e?

Yes it could: Be fracture. You could also have compartment syndrome. Show it to a doctor for proper treatment to avoid permanent damage.

Swelling can be: Any sign of fracture or bad bruising or bleeding or formation of a hematoma or crush injury or infection or tendinitis . Fracture is highly up on the list for swelling after an injury. The way one is evaluated to see if there is a fracture is to have an exam and then get x-rays if the examiner feels their needed. Whether this is done in a cute emergency room setting or a doctors office depends upon the acuteness of the situation and many other factors but often it it's been a few days there's no longer need to rush to the emergency room but an evaluation sounds like a good idea.