I agree with mgcrules that wreck sounds more significant in damage than accident. The two words are very different in meaning, but not as regards BE and AE. I could be wrong. The British folks will be arising for breakfast very soon and will answer more definitively.
A car crash or a car accident (or motor accident, road accident, traffic accident). A crash leaves really visible damage: a crunched panel or a broken light at least. If it just left a scratch or took off a bit of paint you wouldn't say crash.
There is a colloquialism 'train wreck', which is probably AE as much as BE, which means a really horrible situation - it could be anything, two people arguing furiously in an office - where you can see it's going to be very bad and messy to deal with afterwards. I don't think real railway accidents would normally be called train wrecks though.
And if the context is clear that it's to do with cars, we often leave out "car" as well.
If I say to a friend "I saw an accident on my way to work", they would usually imagine a car accident of some kind. If it were a serious one, I'd probably say "I saw a crash on my way to work" they would know it was a serious one.
Other more colloquial words include "prang" as in "I pranged my car" or "I had a bit of a prang" (meaning minor damage, more embarrassing than dangerous).