Thanks for answering. 9 lines of code down to 2.
So much to learn about using this tool properly!

You could use:

i = 0;
Map[{++i, #} &, allVecs, {2}]

or

i = 0;
Replace[allVecs, x_ :> {++i, x}, {2}]

or

Internal`PartitionRagged[MapIndexed[{#2[[1]], #1} &, Join @@ allVecs], Length /@ allVecs]

or

tmp = Join @@ allVecs
tmp = Transpose[{Range[Length[tmp]], tmp}];
Internal`PartitionRagged[tmp, Length /@ allVecs]

or

tmp = Join @@ allVecs
tmp = Transpose[{Range[Length[tmp]], tmp}];
lens = Length /@ allVecs;
begins = Accumulate[Prepend[Most[lens], 1]];
ends = Accumulate[lens];
MapThread[tmp[[#1 ;; #2]] &, {begins, ends}]

or

pos = Position[allVecs, _, {2}, Heads -> False]
numbers = Range[Length[pos]]
replacements = MapThread[#1 -> {#2, Extract[allVecs, #1]} &, {pos, numbers}]
ReplacePart[allVecs, replacements]