Um the idea of a pendulum in an old fashioned clock is that it is actually the clock's frequency reference. It's purely mechanical, no electricity or radios. The length of the pendulum determines the frequency (usually 1 hz). You can slide the weight up and down a little bit to adjust the speed. The spring unwinding gives the pendulum a little kick on every swing so the clock doesn't stop. You wind up the spring every so often so it doesn't unwind completely, and the swinging pendulum advances a little ratchet that moves the hands a little on every swing. If you lived in a town in the pre-electricity era, the local church would ring its church bells at noon, 3pm, etc. and you would use that to set or adjust your clock as needed. The church clock itself was directly or indirectly set using solar noon (as observed with a transit telescope or dipleidoscope) as a reference. Fancier pendulum clocks had various sorts of thermal compensation and could be very accurate. It was a highly developed technology that is now mostly forgotten.

Connecting wifi to this would be at best purely decorative. I guess it would be a cute hack but meh. You could look on hackaday.com which is full of projects like that. I've mostly found them kind of pointless, but that's just me being a grouch.