A very warm welcome to the CameraLabs forum.
I get a sense that there isn't a compact camera out there that matches your specification but I would be delighted to be proved wrong. For instance, the Ricoh Caplio GX100 has a timer which runs from one frame every 5 seconds to one every 3 hours but, searching the net this afternoon, I couldn't find one that goes faster. Indeed, some compact cameras max out at 1 fps continuous shooting.
Canon produced the Digital IXUS Wireless
compact which can be remote controlled from a PC. I think the camera can still be found but I can't imagine that this is the solution you want. If you are prepared to lug a notebook around then why not use a DSLR? In any event, I imagine camera battery life would be a real issue.
Maybe it would help forum members more knowledgeable than myself if you were to share your intended application(s) and budget and why it has to be a compact camera.
Sorry not to be able to solve your problem.