How to drive 16 stepper motors with Arduino?
If you need to drive all 16 simultaneously and independently, the Uno may not have enough capability to do it. Consider the worst-case number of motors and highest rate(s) you expect to drive at once, plus anything else you need the Uno doing at the same time. You might try a simple sketch to do all of these things (substituting LEDs for the motor drivers) at once, and use a scope to examine those LED output signals for the right frequency, wave-form, and lack of jitter. Without I/O expanders, you won't get to 16 motors' worth of outputs (unless you already have an I/O expander) but you might try doubling up on the existing output (use non-overlapping motor phases if you can).

If the Uno can manage that with clean outputs, you'll have a better idea whether it can manage the full load. If it doesn't, you may need to reconsider your choice of processor with respect to its clock-speed, both kinds of memory (depending on what you learned), number of outputs, possibly number of available timers and of interrupts.