Muncey Construction
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Now I could list the steps that I think Muncey had used.

  1. He builds a square of normal integer that look like an snake:
              1  2  3 .. 12
          24 23 22   13
          25 ..
  2. He changes some values from first line to the last line, like this:
            1  143 142 141 140 139 7  8   ...
        ..
       144  2   3   4   5   6 138 137 ...
  3. He changes some values from second line to the eleventh line
  4. He goes on with the other lines. The rule is to make the magic constant of rows near to the 870 value, since the columns are already in that situation (because of the snake construction)
  5. He converts this square to the corresponding prime square
  6. He makes one row to the magic constant changing values of two (or more) lines (the value are taken from the same columns for not changing columns state)
  7. The previous step is repeated for all the lines
  8. He makes one columns to the magic constant changing values of two (or more) columns (the value are taken from the same rows for not changing columns state)
  9. The previous step is repeated for all the columns
  10. He makes one diagonal to the magic constant changing some values, so some columns or rows are change they magic state
  11. He makes the other diagonal to the magic constant like in the previous step
  12. He makes the rows or columns to the right magic state changing values in the same rows (or columns) that are not in the diagonals (for not changing their values)
  13. Points 11 and 12 can be inverted and then using another point like the actual 12
Ok, this seems to be likely, or no?
No. Order 12 is less complicated that the 15 and 17 that I had worked manually on, so the previous points will be not all present, or they are in other order.
Why do we try to build the Muncey square manually? Ok, go.

Note: even if Muncey works on prime square, I use to shows his probably shifts in the corresponding non prime square (because I use refer to integer sub sequential number, so there's less probably that I report invalid shift numbers by typing them).

First Step

Now we build the snake square and then we build the corresponding prime snake square: 

      1   2   3   4   5   6   7   8   9  10  11  12    792
     24  23  22  21  20  19  18  17  16  15  14  13    648
     25  26  27  28  29  30  31  32  33  34  35  36    504
     48  47  46  45  44  43  42  41  40  39  38  37    360
     49  50  51  52  53  54  55  56  57  58  59  60    216
     72  71  70  69  68  67  66  65  64  63  62  61    72
     73  74  75  76  77  78  79  80  81  82  83  84   -72
     96  95  94  93  92  91  90  89  88  87  86  85   -216
     97  98  99 100 101 102 103 104 105 106 107 108   -360
    120 119 118 117 116 115 114 113 112 111 110 109   -504
    121 122 123 124 125 126 127 128 129 130 131 132   -648
    144 143 142 141 140 139 138 137 136 135 134 133   -792

-6   0   0   0   0   0   0   0   0   0   0   0   0      6

      1   3   5   7  11  13  17  19  23  29  31  37   4318
     89  83  79  73  71  67  61  59  53  47  43  41   3748
     97 101 103 107 109 113 127 131 137 139 149 151   3050
    223 211 199 197 193 191 181 179 173 167 163 157   2280
    227 229 233 239 241 251 257 263 269 271 277 281   1476
    359 353 349 347 337 331 317 313 311 307 293 283   614
    367 373 379 383 389 397 401 409 419 421 431 433  -288
    503 499 491 487 479 467 463 461 457 449 443 439  -1124
    509 521 523 541 547 557 563 569 571 577 587 593  -2144
    659 653 647 643 641 631 619 617 613 607 601 599  -3016
    661 673 677 683 691 701 709 719 727 733 739 743  -3942
    827 823 821 811 809 797 787 773 769 761 757 751  -4972

-22  -8  -8   8  -4  -4  -2  12   2  -8   6   0  6    28

Second Step

Since the columns are almost near to the magic, we suppose that Muncey has decided to systematize them from immediately. Now I list the shift that behavior needs for systematize it departing from the first and going on. 

rows:             columns 
 5    49 <->  50     1
 6    72 <->  70     1

 4    46 <->  47     2 
 6    71 <->  72     2
 7    75 <->  74     2
 8    94 <->  95     2
 9    99 <->  98     2

 1     3 <->   4     3
 2    22 <->  21     3
10   117 <-> 118     3

 6    69 <->  68     4
12   141 <-> 140     4

1°:    5 <->   6     5
8°:   92 <->  89     5
Note that now one diagonal and columns 6 are become magic. 
 1     7 <->   8     7
 7    79 <->  82     7
 8    90 <->  86     7
11   127 <-> 128     7

 2    17 <->  10 *   8
 4    15 <->  41 *   8
 6    65 <->  64 
 7    80 <->  90 *   8
10   113 <-> 129 *   8
12   136 <-> 137

 1     9 <->  17     9
 2    41 <->  16     9
 7    81 <->  79     9
 8    88 <->  80     9
 9   106 <-> 105     9
10   111 <-> 112     9

 1     9 <->  11    10
 8    87 <->  85    10
11   130 <-> 132    10

11   130 <-> 131    11
To make the other diagonal ok, we must do this: 
      90 <-> 92 *
Note that the * means that the shift was not done in the same row. The strange thing is that the number 90 was changed two time with *. This probably means that Muncey first makes the diagonals to the right state, and then changes the columns state: 
      1   2   3   4   5   6   7   8   9  10  11  12    792
     24  23  22  21  20  19  18  17  16  15  14  13    648
     25  26  27  28  29  30  31  32  33  34  35  36    504
     48  47  46  45  44  43  42  41  40  39  38  37    360
     49  50  51  52  53  54  55  56  57  58  59  60    216
     72  71  70  69  68  67  66  65  64  63  62  61    72
     73  74  75  76  77  78  79  80  81  82  83  84   -72
     96  95  94  93  92  91  90  89  88  87  86  85   -216
     97  98  99 100 101 102 103 104 105 106 107 108   -360
    120 119 118 117 116 115 114 113 112 111 110 109   -504
    121 122 123 124 125 126 127 128 129 130 131 132   -648
    144 143 142 141 140 139 138 137 136 135 134 133   -792

-6   0   0   0   0   0   0   0   0   0   0   0   0      6

      1   3   5   7  11  13  17  19  23  29  31  37   4318
     89  83  79  73  71  67  61  59  53  47  43  41   3748
     97 101 103 107 109 113 127 131 137 139 149 151   3050
    223 211 199 197 193 191 181 179 173 167 163 157   2280
    227 229 233 239 241 251 257 263 269 271 277 281   1476
    359 353 349 347 337 331 317 313 311 307 293 283   614
    367 373 379 383 389 397 401 409 419 421 431 433  -288
    503 499 491 487 479 467 463 461 457 449 443 439  -1124
    509 521 523 541 547 557 563 569 571 577 587 593  -2144
    659 653 647 643 641 631 619 617 613 607 601 599  -3016
    661 673 677 683 691 701 709 719 727 733 739 743  -3942
    827 823 821 811 809 797 787 773 769 761 757 751  -4972

-22  -8  -8   8  -4  -4  -2  12   2  -8  6   0   6    28
In the previous squares, the right values for the diagonal are in bold, instead, the same color are used for the values that are to be swapped.
For the other diagonal: 
      1   2   3   4   5   6   7   8   9  10  11  12   792
     24  23  22  21  20  19  18  17  16  15  14  13   648
     25  26  27  28  29  30  31  32  33  34  35  36   504
     48  47  46  45  44  43  42  41  40  39  38  37   360
     49  50  51  52  53  54  55  56  57  58  59  60   216
     72  71  70  69  68  67  66  65  64  63  62  61   72
     73  74  75  76  77  78  79  80  81  82  83  84  -72
     96  95  94  93  89  91  90  92  88  87  86  85  -216
     97  98  99 100 101 102 103 104 105 106 107 108  -360
    120 119 117 118 116 115 114 113 112 111 110 109  -504
    121 122 123 124 125 126 127 128 129 130 131 132  -648
    144 143 142 141 140 139 138 137 136 135 134 133  -792

 -2  0   0   1  -1   3   0   0  -3   0   0   0   0     3

      1   3   5   7  11  13  17  19  23  29  31  37   4318
     89  83  79  73  71  67  61  59  53  47  43  41   3748
     97 101 103 107 109 113 127 131 137 139 149 151   3050
    223 211 199 197 193 191 181 179 173 167 163 157   2280
    227 229 233 239 241 251 257 263 269 271 277 281   1476
    359 353 349 347 337 331 317 313 311 307 293 283   614
    367 373 379 383 389 397 401 409 419 421 431 433  -288
    503 499 491 487 461 467 463 479 457 449 443 439  -1124
    509 521 523 541 547 557 563 569 571 577 587 593  -2144
    659 653 643 647 641 631 619 617 613 607 601 599  -3016
    661 673 677 683 691 701 709 719 727 733 739 743  -3942
    827 823 821 811 809 797 787 773 769 761 757 751  -4972

  0 -8  -8  12  -8  14  -2  12  -16 -8   6   0   6    10
So, we obtain: 
      1   3   5   7  11  13  17  19  23  29  31  37   4318
     89  83  79  73  71  67  61  59  53  47  43  41   3748
     97 101 103 107 109 113 127 131 137 139 149 151   3050
    223 211 199 197 193 191 181 179 173 167 163 157   2280
    227 229 233 239 241 251 257 263 269 271 277 281   1476
    359 353 349 347 337 331 317 313 311 307 293 283   614
    367 373 379 383 389 397 421 409 419 401 431 433  -288
    503 499 491 487 461 467 479 463 457 449 443 439  -1124
    509 521 523 541 547 557 563 569 577 571 587 593  -2144
    659 653 643 647 641 631 619 617 613 607 601 599  -3016
    661 673 677 683 691 701 709 719 727 733 739 743  -3942
    827 823 821 811 809 797 787 773 769 761 757 751  -4972
  0 -8  -8  12  -8  14  -2  -24  0  -14  32  0   6     0
The above changes are for making the first row to the magic state. In fact, I must suppose that now Muncey makes first the row, and finally the columns. 
      1   3   5   7  11  13  17  19  23  29  31  37   4318
     89  83  79  73  71  67  61  59  53  47  43  41   3748
     97 101 103 107 109 113 127 131 137 139 149 151   3050
    223 211 199 197 193 191 181 179 173 167 163 157   2280
    229 227 233 239 241 251 257 263 269 271 277 281   1476
    349 353 359 347 337 331 317 313 311 307 293 283   614
    367 373 379 383 389 397 421 409 419 401 431 433  -288
    503 499 491 487 461 467 479 463 457 449 443 439  -1124
    509 521 523 541 547 557 563 569 577 571 587 593  -2144
    659 653 643 647 641 631 619 617 613 607 601 599  -3016
    661 673 677 683 691 701 709 719 727 733 739 743  -3942
    827 823 821 811 809 797 787 773 769 761 757 751  -4972
 0   0  -6   2  -8  14  -2  -24  0  -14  32  0   6    0
The above are the changes for making the row 2 to the magic state. The strange thing is that if you only make red and blue changes, you obtain the magic state. So, why use another 3 changes?.

This learn us that is very difficult this reverse engineering about the Muncey works. On the other hand, this has shows us that even if we don't use computer (like in 1913) the order 12 is easy to make magic if we use an initial construction similar to the snake (at this point you must suppose that Muncey may have use different initial construction, but I think that the snake construction is the more probably), and make first the diagonals to the magic state. Final note: may programs of order 12 never find solution if we first make the diagonals magic, so this means that we must change values from point that are not in the same columns and rows for making them magic.

 
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