How can I monitor AC current in multiple wires with a single ammeter?
Multiplexing the actual AC current is not a good idea.  If you're going to multiplex anything, it should be the small current signals.

That means you are going to have a sensor per AC line.  The simplest is probably to use dedicated current transformers.  The clip on transformers are for convenience, but more expensive and larger than dedicated fixed-mounted current transformers.  You'll have to do a little re-wiring, but only once.  I would include the burden resistors with each transformer.

Once you have the two current transformer output wires for each AC line you want to measure, you have more options.  One possibility is to have these all come to one board that does the absolute value and low pass filter functions on each signal, and has a microcontroller scan the results.  You adjust the low pass filters so that the micro can scan each signal without aliasing.  The micro then reports all the aggregated measurements to wherever you want them, or displays them, or whatever.  You didn't say what you wanted to do with the results.

However, if the purpose is to measure power, then you really need to take the waveforms into account.  Lots of devices don't look resistive and can have significant power factors.  Put another way:

&nbsp;&nbsp;Watts = ave(volts x amps)

The product of average volts and average amps is NOT power, and has little meaning.  There is something called the "VA" (Volts x Amps) measurement:

&nbsp;&nbsp;VA = rms(volts) x rms(amps)

but that's not real power either, and even this requires RMS volts and amps, not just average.  The ratio of Watts to VA is the power factor.  Only a pure resistive load has a power factor of 1 such that you can get away with computing VA and using it as real power.

If you are trying to measure power consumption, like what the power company bills you for, you want real power, not VA.  This is not so simple to measure because you have to sample the voltage and current many times per power line cycle, take the product of each of the sample pairs, and low pass filter the result.  Multiplexing all that would be difficult, so it effectively means a microcontroller per AC line.  These can then all dump out their data over a CAN bus or something, which allows collecting it all in one place.