The problem is not really about products, more about finding an arrangement of digits 1 to N-1 (for base N, since digit N does not exist in base N), that conforms to the criterion that any pair in the arrangement is part of the set of single digit products.
There could be a better way to solve this, but I wrote a small program that creates a table of possible digit combinations and uses that to try to find a valid arrangement.
It's not optimal in terms of performance (essentially a depth-first-search), but finding a pattern for base-N takes N times longer than for base-N-1.
this My program was able to show that bases 3, 4, 5, and 6 do not have such an arrangement (due to digits that do not exist in a product).
Interestingly, for base-2 (binary) it thinks that "1" is a valid combination due to the only non-zero digit being 1, so the row has no neighbouring digits.
Otherwise it shows that bases 7 to 21 have possible arrangements of digits.
base-2: 1 (questionable)
base-3: -
base-4: -
base-5: -
base-6: -
base-7: 513426
base-8: 5243617
base-9: 46271538
base-10: 728163549
base-11: 739158264a
base-12: 384765a192b
base-13: 35786a294b1c
base-14: 385b27a496c1d
base-15: 375d1b4a698c2e
base-16: 375b6e19c4d2a8f
base-17: 35b8e2ac6f17d49g
base-18: 397af2d6bc5e48g1h
base-19: 37bca59g2d8h1f4e6i
base-20: 35h28be97cf6dag4i1j
Here's a link to a program you can run in your browser.