You may want to define the term "real expensive"!
Canon makes some very nice camera's with UW housings. I would recommend that anything you look into has a housing available. If you want to go on the cheap, get a housing built by the manufacture vs. a third party housing. While the third party housings are generally better, they are also generally at least 2X the cost.
The Canon powershot G10 would be a great choice, but maybe a bit pricey, and not super compact. They also have S series and A series cameras that may have UW housings available. Those tend to have less features vs. the G series, but are more compact.
Nikon also makes some nice compact cameras. Again, check on housing availability and features before purchase.
IMO a good UW compact camera will have the ability to shoot in Manual mode, RAW is nice, and have a lens that is 28mm wide. Shutter lag is a big factor in these cameras, but maybe not something one has a lot of say in especially on a budget.
Most compact manufacture housings fall in the $200~$300 range, and the camera's run anywhere from $125~$450+. Ikelite housing generally run in the $500+ range for compacts. You may want to read about what other users are shooting on SB as a starting point.
Unfortunately without an UW flash, your shooting possibilities are somewhat limited. However my Avatar was shot with a Canon S70 in a Canon housing without a flash, so if you are doing shallow dives on sunny day good results are very attainable. Post processing is a MUST for most UW shots especially if shot without a flash. Software tools is an entirely different topic of discussion.
You may want to look at used equipment on Ebay if budget is a factor. Make sure you can find both the housing and camera if you are purchasing used. It's difficult to find new housings for discontinued camera's, and versa visa.