Now days diving contractors will frequently gear up for a specific job if it's profitible enough, lease gear and hire divers with the certs rather than train up internally. Check out the AWS website, training etc.
Not all company will except AWS certs for welding qualifications some require you to weld a test sample, any structural steel (including bridge work) in NYC you need to take a test from the city to get license and in CT you need to get a "G" license for pressure vessles and pipe.
There is alot more underwater burning than welding mostly for salvage. My self and most of the guy I know who do underwater welding it is mostly patch work not so much constuction. One good literature resorce for underwater welding is the US Navy underwater welding guide.
I personally took a top side 900 hour welder fitter course @ W.T.T.I. of Allentown Pa. years back which has been one of the best things I have ever done, great school, good time & I actually work with them now. I also attended IDI in Charleston, mostly just for the fun of it, also a great time! I am currently getting in to the various realms of inspection and may possibly toy with the idea of taking some of that under water at some point. If you like constant travel and bubbling around be a diver but know that wet welding will never be your sole purpose as one. Habitat welding is cool as balls but I agree you must be hot s*** for it. If you have the means, I would recommend taking the same top side 900 hour (roughly 9 months) course I did, you will never regret it. Then simply get the diving as another tool to your belt. All in all the time and financial commitment's are not that huge relatively speaking. You will have lots of choices either way then.