Educated Guesswork

I have a FujiFilm MX2900 zoom, and the same complaint as you. I thought it was just because it's an older camera, but I guess that's not the case? I know that the time it takes to store the image is a function of data transfer speed, but I wonder what causes the massive shutter delay...

In my experience, this is often due to the need for the flash. It takes a while for the capacitor to charge and if you have taken a picture recently, it's not fully charged when you hit the button. It seems to be worse on my camera the lower the batteries are. An easy way to test this is to go outside on a sunny day and see if the same delay is there.

It's not just the flash, it's the photo sensor array. Your Nikon SLR probably had a CMOS sensor like my canon D30, where most cheaper compact cameras generally use CCD sensors. More details here: http://www.imaging-resource.com/PRODS/D30/D30A4.HTM

My Nikon SLR had a silver-nitrate-based emulsion sensor laid down on a roll of plastic :)

Could it be the autofocus taking a while to compute?

It's not a CCD vs CMOS thing. It's mostly the auto-focus plus the time it take the camera to come out of it's sleep mode. Google for "shutter lag time". The auto-focus lag is time from start of 1/2 press to point you have focus. Then you have the shutter release lag which is time from 1/2 press to taking the shot. If you just go for a full press, you get total lag. Most point and shoot things suck for photographing anything moving faster than an IETF draft. This has become a hot topic in low end cameras and some of the new small cameras are getting better. Mostly you need to move to a SLR form factor if you don't want silly shutter lag.