Nearly every cell phone camera uses an electronic shutter rather then a mechanical shutter. Once that is understood, the rest of the equation is simply how fast the software capture the shot.
You stated that you have had the Apple iPhone 4 and considered it laggy. The iPhone 4 is on a path to become the most used camera on Flickr, and I think many people would consider it to be one of the better camera phone options.
The easiest way to work around its "laggy" shutter release is to use a variety of third party applications available in the app store. My favorite is Camera+ which dramatically speeds up image taking abilities as compared to the built in Apple Camera application. They do this by storing photos into a temporary lightbox management tool within the application. This allows you to take burst shots of as many images as you would like, extremely quickly.
I have found that the speed of the burst shots taken with this application rival many DSLRs of today. The one thing you will want to understand about this is that when they are saved to this lightbox within the application, they are not available in the standard "Photos" application till you export them from the Camera+ lightbox. Some people do not like this piece of it, but I think it is a benefit that allows you to only export and save the images that you liked the best, especially out of a burst of shots.
With the release of iOS 5, Apple claims to greatly enhance the photo taking capabilities of the current generation of iPhone 4s and iPod Touches. It not only speeds up the process of taking photos using the software enhancements, but it also gives the user the ability to open the camera application directly from the lock screen of the phone, as well as use physical the volume button to actually activate the electronic shutter.
Furthermore, if you are trying to take images of small children while moving around, a camera phone may not be the best possible solution. You may be experiencing quite a bit of motion blur due to the children moving around. The majority of the current generation of camera phones simply do not have large enough apertures or high enough ISO capabilities to freeze the action of small children. A DSLR with a 50mm f/1.8 lens might be the best solution for images like this!