If I buy one of these will it take a picture (assuming it's had time to focus) the 'instant' I've pushed the shutter and not half a second later!

When photographing kids the moment can be missed if the shutter doesn't fire instantly!

Almost every camera you use will take a picture half a second later, unless you bump up to an SLR. You can try to manually override the automatic settings to lower the shutter speed, and raise the ISO settings, causing the shutter to open and close quickly, again harder in a point and shoot than an SLR. One thing that I would recommend is the pre-focus feature that every camera has: When you push the capture button down halfway it causes the camera to pre-focus, taking a picture immediately. Or, the other thing you could try is shooting in burst mode, which takes several photos in a short amount of time, and you can pick and choose the ones you want to come. In most instances I would recommend the Sony H9 for this over the cannon because the settings are easier to manipulate, plus it has a one touch burst mode button so you don't have to switch settings when there's lots of action going on. If you wanted to stay with a smaller point and shoot, casio has the fastest focus time, with panasonic as a close second, followed by cannon and nikon, then finally sony for the lower end cameras.