Welcome to the Forum!

Usually, people who enter the world of underwater welding, have some skills in welding, but have held an interest in diving.  Not saying you can't do it the other way around, but I suspect that leaning welding from scratch is more difficult that learning to dive.

I cannot comment on the trade schools (don't know where you reside).  Good luck, and hope it turns out.

Tyrone

Could you, yes anyone can.
Should you is the important question.
16 years ago I did the same thing your considering doing .  I left my full time job becasue I was bored and went to The College of Oceaneering in Long Beach CA for a 1 yr program to be an underwater welder.  Once I finished the first 7 months of diving training, then we move onto the speciality training ie welding, inspection, or medical at that school.  The thing is basically anyone can dive but what can you do once you get down there? You need a trade, mechanic, valves, weld, inspect, look for pipe, whatever.

Once I graduated and went to the gulf I worked offshore.  You won't dive at first you'll be a red hat tender which means your a grunt and a gopher and your the low man on the pole.  After while you can work yourself up to list and strat diving on little jobs here and there.

When I was in school I saw what the welders did for practise which was run pass after pass after pass all day every day.  I personally couldnt do that without going insane and getting bored.  Now I know it takes practise to get the technique down, but I didn't want to do that for 4-6 weeks, that's just me.
I saw the inspection guys setting up underwater cameras, video, MT units and the like and thought that was more what I wanted to do so I took the inspection program instead.  I haven't looked back and it took off for me and worked out great.  Ive been a CWI for 15 yrs now and it's been great I cound'nt be better off.

Now I don't dive, in fact I left the gulf after about 6 months for personnal reasons but the training I got was a great step forward.  But be careful is the cost going to offset the benefit? only you can answer that question.  It's not for everyone and its not glamourous and its hard work away from home and if/ when the weather is bad they dive anyway. That's the commercial part they don't care if its raining out or snowing, you'll be standing on the deck of some boat for 2 hrs holding a hose letting it out or taking upthe slack. 
Do your homework and make sure this is really for you.
Good luck

Ladies and Gentlemen:
       It is spring and thoughts go to going down to the sea and under it. I got my dive school diploma at 30 years old as an experienced engineer in 1975. I graduated from the same diving school as the other respondent and am CWI qualified. . I have an extensive underwater welding background how be it not current being a graduate of a marine diving school http://natpoly.edu Wilmington California and worked in the management  of the company with world's largest diving contract with world largest oil producing company. As a career move, I would suggest giving it thought. Look in the newspaper or other help wanted places and count the number of fits for welders and/or welding technicians compared to marine divers. A hundred to one away from marine diving is a guess. You might spend time contacting marine diving concerns and see their response to you looking for the work. The work is seasonal. If you put ten thousand on the course, does the course you have a shot at New York area dock builder diving work. I am part of LU 1556 ( local New York union in marine construction etc.) . As foreman again you would not be my top pick for diving.  Personally I did underwater welding, invented underwater technology ( sticks ) resulting in five patent, went through 20 -30 wet welders on one job and was deemed an expert in underwater welding by the  IIW ( International Institute of Welding ) and had fun with it. My background includes ten patent five in welding and a current employment as an engineers and having been a merchant seaman. Today there was talk of me doing some underwater welding. I hope this is helpful.
Sincerely
Len Andersen    www.lenandersen.com
914-536-7101 (Mobil) / 212-839-6599, 4042 FAX (o)/ 914-237-7689 (H) / 
www.lenandersen.com