Need faster analog mux. Use switches and summation amp?
Just some guesses since you haven't given us enough information:

Analog MUXes are usually just FETs; you can create your own MUXes by selecting FETs with fast switching characteristics and matching by hand, or by using arrays of FETs which are generally very well matched already. Since you haven't given us the MUX you're using or provided a schematic I can't really suggest anything specific. You also haven't given us the source signal range so we can't even verify that your MUX selection is appropriate or offer suggestions.

The time it takes for the ADC s/h circuit to charge to the new signal level is a function of the source impedance. If your sources are relatively high impedance it is going to take longer to sample, and since you're rapidly switching between inputs you want the source impedance to be as low as is reasonable (and s/h capacitor to be small and have a low impedance as well.)  You can help the former by using fast op-amps configured for unity-gain (voltage followers) feeding the MUX, and you may need another one after the MUX depending on the specific characteristics of the MUX. Unfortunately you haven't given us the ADC you're using so we can't help you by verifying or commenting on the appropriateness of it or offering specific suggestions for speeding your sampling system up.

You may want to edit your question, include the missing information I mentioned here, adding a (readable!) schematic and following all the general guidelines you see mentioned here time and again on writing good questions. Without this, we can only offer general suggestions.