Don't know how cheap you want to do such a trip, but you could hitch a lift on a container ship? No joke, some people have done it and for a small fee and a small bunk, just offer to do little jobs on board, make teas, sweep floors, service the crew. (Okay I'm just kidding about the last one.) Email a cargo company and ask if they'll consider it. They leave from south coast ports, or more likely for an Atlantic trip, Liverpool.
I had a mate looking into a few years ago because he was so scared of flying. In the end he didn't fancy the idea of a fortnight at sea, and decided to just take Valium, get a bit drunk, and take a transatlantic flight. True story.