Not sure about a "mechanical" solution, but if you want to do it with paper and pencil...
I did it with paper and pencil. And no conversion from hexadecimal to binary was needed, except rotations. But I know that mechanically it should be possible. When it comes to addition, it is possible to create "addition table", like this:
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| + | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | a | b | c | d | e | f |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 0 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | a | b | c | d | e | f |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | a | b | c | d | e | f | 0 |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 2 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | a | b | c | d | e | f | 0 | 1 |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 3 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | a | b | c | d | e | f | 0 | 1 | 2 |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 4 | 4 | 5 | 6 | 7 | 8 | 9 | a | b | c | d | e | f | 0 | 1 | 2 | 3 |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 5 | 5 | 6 | 7 | 8 | 9 | a | b | c | d | e | f | 0 | 1 | 2 | 3 | 4 |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 6 | 6 | 7 | 8 | 9 | a | b | c | d | e | f | 0 | 1 | 2 | 3 | 4 | 5 |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 7 | 7 | 8 | 9 | a | b | c | d | e | f | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 8 | 8 | 9 | a | b | c | d | e | f | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 9 | 9 | a | b | c | d | e | f | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| a | a | b | c | d | e | f | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| b | b | c | d | e | f | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | a |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| c | c | d | e | f | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | a | b |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| d | d | e | f | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | a | b | c |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| e | e | f | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | a | b | c | d |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| f | f | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | a | b | c | d | e |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
I created more such tables, for moving values during addition, multiplication (needed to validate k-values), and other things like that. After few hours, I could add hexadecimal numbers as well as decimal ones. Constructing similar tables for rotations is also possible, but it is harder. So yes, I know how to do it by hand, and I can reach a bit better hashrate than on this video, but it is still not sufficient. I think about something similar to a
mechanical calculator, but it should work on SHA-256 (or even better: should also allow hashing things multiple times), should use hexadecimal (or binary, or other power-of-two-based system) keyboard, and allow quite fast calculation, by setting IV, setting data, and then it should turn Initialization Vector into Exit Hash. It could work on-the-fly, and modify w-values on-the-fly, then going backwards could be possible by rotating it mechanically in the opposite direction.
