In general, a helpful strategy is to list out the possible entries for each clue in several bases to see what lines up. Some additional helpful pointers:
5A must be either 020 or 100, so that gives you one digit right away
The factorial clue must be a small base, since the number of digits in a factorial increases rapidly. This is enough in conjunction with 5A to solve both clues.
The value of 3D is known since it’s only single digits in the clue (decimal 216); it must be at least base 7, or else it has too many digits
Similarly, 4A is guaranteed to be divisible by 3, which helps to narrow down the base
1D is actually quite constraining; once 5A is solved, it should confirm this base as well
Thanks everyone for solving! This was fun to make and I hope to do more like this sometime!