Here's a very lightly tested attempt:

ConcatenateCsvFiles[output_String, inputs : {__String}] :=
 With[
  {joinedData = Import /@ inputs},
  Export[output, Prepend[Flatten[joinedData[[All, 2 ;;]], 1], joinedData[[1, 1]]]]]

But I should make some comments. First, I probably wouldn't actually do all three steps in one big function like this. I would probably do the Import and Export separately from the catenation of the list data. It would be easier to test and parts could be re-used independently. It also reduces the risk of overwriting good data with bad.

Second, this doesn't match what you asked for:

contatenate[singleFilename,#]& /@ {f1.csv,f2.csv,....fn.csv}

But what you asked for would probably need to rely on updating some temporary state, and it would just become awkward. The semantics of the problem (as I understand it) isn't about applying the same function to a list of data, but about restructuring a list of (lists of) data.

Third, if you're never going to process the data itself--i.e. this is strictly a file-to-file convenience tool, then there might be a neater approach that doesn't bother with importing structured data but just processes the files' contents as just lines.

Join works with Dataset

ds1 = Dataset[{<|"a" -> 1, "b" -> "x"|>, <|"a" -> 2, "b" -> "y"|>}]
ds2 = Dataset[{<|"a" -> 3, "b" -> "z"|>, <|"a" -> 4, "b" -> "t"|>}]
ds1~Join~ds2

There might be a nifty way to do it with datasets directly, but the way that I immediately thought of was to use Normal to "unwrap" the dataset and then apply Dataset again. Let's say you have data1, data2, and data3 that are Dataset expressions that you have already imported from the csv files. You could do this:

Dataset[Normal@{data1, data2, data3}]

Normal turns the datasets into lists of Associations. Dataset automatically works with such Associations. You might have to Flatten the "normalized" data in some circumstances--not sure.

No need for that complexity, as I mentioned earlier, this works fine

t = d1~Join~d2

Great! I didn't realize that you could just join them like that. I assumed that you needed first to get the lists of associations (thus the use of Normal), join the list, and then turn the list into a Dataset.

Your solution is just so much simpler.

Thank you!