Any recent models will not have that problem. Basically, you don't really have to worry about this aspect when you choose your new camera. There are still differences in timing for different models, but not really significant given that you'll take time to compose the next picture. A lot of cameras also offer continuous mode shooting, which will be the answer if you need that function.
Canon always produces reliable cameras (not necessarily sleek to look at). For your price range, the PowerShot A series should suit your needs (like the A80). The PowerShot S series gives you more control; more expensive, but you might find a model within your budget.
Olympus cameras are actually very good too.
A few examples of cycle time (between shots) reported on click here :
Canon PowerShot A80 is less than 1.5s.
PowerShot S50 is about 1.7s.
Olympus Stylus (or mju) 400 (they have different model names in different countries) - about 2.4s for larger higher quality photos, and 1.9s for smaller ones.
Olympus C-50 Zoom is about 2.8s, the c-750 Ultra Zoom is 2 to 2.5s.
Nikon Coolpix 4300 is about 2.5s. 3700 is 1.5 to 2s.
All the timings are fast enough for practical purposes.
So by all means check the cycle times in reviews (they never form part of the camera specifications - unless the time is really fast), but I suggest you concentrate on other specifications, such as lens range, pixel count and other functions.
One thing you'll have to ditch is the SmartMedia card. Olympus used to use them but have given up on them.