Stop Canonical Ordering of Output List

alternatives = N19 | N560 | N41 | N61 | N551 | N20 | N539;

You can map Cases over the list of alternatives:

Join @@ Cases[{#, __}][m] & /@ (List @@ alternatives);

TableForm @ %

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Alternatively, sort the output of Cases by the position in alternatives of the first column elements:

SortBy[PositionIndex[List @@ alternatives]@*First][Cases[m, {alternatives, __}]];

% // TableForm

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The cases will be found in the order they occur in m. If each case is unique, as in the example, the following can be used:

Sort[
 Cases[m, {N19 | N560 | N41 | N61 | N551 | N20 | N539, __}]
 ][[Ordering[N19 | N560 | N41 | N61 | N551 | N20 | N539]]]

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