Using PositionIndex and Merge,

mergeChron[x_] := Merge[PositionIndex /@ x, Max] // Sort // Keys;

Test:

Case1:

allsublists= {{a1, a5}, {a3, a2, a5}, {a1, a2}};
mergeChron[allsublists]

{a1, a3, a2, a5}

Case2:

alist = RandomSample[Table[ToExpression["a" <> ToString[i]], {i, 1, 1000}]];
range = Range[1000];
Do[len = RandomInteger[{450, 1000}];
 select = Sort[RandomSample[range, len]];
 sublists[j] = alist[[select]];, {j, 1, 300}]

For num = 300, all three of the methods produce the same output. However, that's not the case for num = 30

num = 300;
allsublists = Array[sublists, num];
f = mergeChron[allsublists];
g = merge[allsublists]; (*Roman's*)
h = topoSort[allsublists]; (*kglr's*)
f == g == h

True

Using PositionIndex and Merge,

mergeChron[x_] := Merge[PositionIndex /@ x, Max] // Sort // Keys;

Test:

Case1:

allsublists= {{a1, a5}, {a3, a2, a5}, {a1, a2}};
mergeChron[allsublists]

{a1, a3, a2, a5}

Case2:

alist = RandomSample[Table[ToExpression["a" <> ToString[i]], {i, 1, 1000}]];
range = Range[1000];
Do[len = RandomInteger[{450, 1000}];
 select = Sort[RandomSample[range, len]];
 sublists[j] = alist[[select]];, {j, 1, 300}]

For num = 300, all three of the methods produce the same output. However, that's not the case for num = 30. It might be because of the existence of multiple solutions (as discussed in comments).

num = 300;
allsublists = Array[sublists, num];
f = mergeChron[allsublists];
g = merge[allsublists]; (*Roman's*)
h = topoSort[allsublists]; (*kglr's*)
f == g == h

True