Is a 1N4001 diode supposed to allow reverse leakage?
I found that there was an odd-looking waveform and it was actually AC
coming out the wrong end of the diode- between +100mV and -100mV to be
exact. It was about 55Hz or so.
Why am I getting an AC voltage out of the wrong side of the diode?

Your scope is picking up EMI.
When the diode is reversed biased it has a very high impedance, so electric fields can easily induce a voltage in the wiring at the 'wrong side'. Your scope probe also has a very high impedance, so it will show EMI that normally would be too weak to see.
The parasitic capacitance coupling the interference into the diode wiring is probably very small, acting as a high pass filter that accentuates rapid changes in the waveform. The peaks in your waveform suggest the original interfering waveform is a square wave, possibly coming from the clock display. Probe the clock display signals. If you see a frequency of ~55 Hz there too then it may be the source.
The current created by this EMI is very small, and since it is AC it will not charge or discharge the battery, so it is nothing to worry about.