How can I figure out an unknown UART's speed?
You want to measure the shortest transition-transition time. 

This will, assuming the bus you're looking at has a decently varied content, tell you the bit-time for the serial interface. You can then go from that to the overall baud rate.



Really though, if you're serious about doing reverse-engineering or any sort of electrical engineering at all, you will need an oscilloscope (and probably a logic analyzer), so you should think about buying one soon.