How to rearrange sublists based upon the element in the first position?

Move D to the front, leaving the order of the rest unchanged:

lis1 /. {a___String, "D", b___String} :> {"D", a, b}

{{{"D", "J", "W", "L"}, {"D", "Q", "R", "T"}, {"D", "O", "P",
"W"}, {"D", "Z", "L", "M"}}}

Swap the first element and D:

lis1 /. {a_String, b___String, "D", c___String} :> {"D", b, a, c}

{{{"D", "J", "W", "L"}, {"D", "R", "T", "Q"}, {"D", "P", "O",
"W"}, {"D", "Z", "L", "M"}}}

If there is a D in every list or if every list should start with D regardless, this would suffice as well:

Map[Prepend["D"], DeleteCases[lis1, "D", {3}], {2}]

{{{"D", "J", "W", "L"}, {"D", "Q", "R", "T"}, {"D", "O", "P",
"W"}, {"D", "Z", "L", "M"}}}

And here's one just for fun:

MapThread[RotateLeft, {lis1, {Position[lis1, "D"][[All, -1]] - 1}}, 2]

{{{"D", "W", "L", "J"}, {"D", "Q", "R", "T"}, {"D", "W", "O",
"P"}, {"D", "Z", "L", "M"}}}

lis2 = {{{"J", "D", "W", "A"}, {"Q", "R", "T", "D"}, 
    {"O", "P", "D",  "W"}, {"D", "Z", "L", "M"}}};

We can also use SortBy or Sort to get the desired results:

If "it doesn't matter what positions the remaining three strings are in":

Map[SortBy[# != "D" &], lis2, {2}]

 {{{"D", "A", "J", "W"}, 
  {"D", "Q", "R", "T"}, 
  {"D", "O", "P", "W"}, 
  {"D", "L", "M", "Z"}}}

If the other elements are to be kept in original order,

Map[SortBy[{# != "D" &, False &}], lis2, {2}]

 {{{"D", "J", "W", "A"}, 
  {"D", "Q", "R", "T"}, 
  {"D", "O", "P", "W"},
  {"D", "Z", "L", "M"}}}

We can use Sort with a custom ordering function:

ord["D", _] = True;
ord[_, "D"] = False;
ord[a_, b_] := Order[a, b];

Map[Sort[#, ord] &, lis2, {2}]

  {{{"D", "A", "J", "W"}, 
  {"D", "Q", "R", "T"}, 
  {"D", "O", "P", "W"}, 
  {"D", "L", "M", "Z"}}}

Use the ord[a_,b_] = 0 to keep the original order of elements other than "D":

Map[Sort[#, ord] &, lis2, {2}]

 {{{"D", "J", "W", "A"}, 
 {"D", "Q", "R", "T"},
 {"D", "O", "P", "W"},
 {"D", "Z", "L", "M"}}}

If "it doesn't matter what positions the remaining three strings are in" and no repeated element in each sublist:

prep[x_] := DeleteDuplicates /@ (Prepend[#, "D"] & /@ First@x)
prep@lis1

{{"D", "J", "W", "L"}, {"D", "Q", "R", "T"}, {"D", "O", "P", "W"}, {"D", "Z", "L", "M"}}

lis2 = First@lis1;
prep[x_] := DeleteDuplicates@Prepend[x, "D"]
prep /@ lis2

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