The wikipedia article on mixed radix numeral systems says
Mixed-radix numbers of the same base can be manipulated using a generalization of manual arithmetic algorithms.
This sounds like "naive long multiplication" in mixed radix systems won't be significantly more complicated than "naive long multiplication" in fixed base systems. However, no algorithm or reference was given, and googling only turned up ridiculously complicated stuff.
Ok, even in fixed base systems, "naive long multiplication" isn't really simple, because one needs the corresponding multiplication table. But can there be something analogous to a multiplication table for mixed radix systems? Or is multiplication in mixed radix numeral systems simply ridiculously complicated?
Is the answer the same for all mixed radix numeral systems, or are some like the factorial number system better suited to handle multiplication than other non-fixed base systems?
It doesn't fully answer your question, but if the radices are repeated (eg. alternating base six and base ten like in Babylonian numerals) then you can generate and use the times tables for base b1xb2x...bn. So in our Babylonian numerals example, every other symbol alternates the bases, so we can just pretend that every pair of symbols is in fact one symbol representing a base sixty numeral. That does mean generating a times table that goes up to sixty times sixty, so it's not very convenient, but it does at least work. As for something that would allow you to multiply in a mixed radix system without reinterpreting it as a single large base (eg. doing Babylonian math with only your tens times tables), I don't know if that's something you can do with a simple algorithm. Something like base factorial would be significantly more complicated and I doubt that a generalised algorithm capable of handling indefinitely long numbers would be manageable for everyday use.