The algorithm for multiplication is pretty easy. If the multiplicand bit is set, add the multiplier to the shift register.

The initial idea was to use a MUX2 to select between the last result (shifted register) or the new result (oh, wait..... nevermind)

But that's typically an electronics-minded design.

Moving backwards, I realise that the multiplier is just a set of manual switches. They are either tied to +Vcc or left open.

Wait.

What if the +Vcc was left open too ? That would add nothing. Which is the desired behaviour for the multiplication by 0. Bingo !

So now, I saved a row of quite a lot of relays, just by routing the multiplicand bit to the +Vcc of the multiplier's rotary hex switches...

(schematics left as an exercise to the reader, or be patient, I must buy groceries...)