I always thought that it would get messy without a mechanism to feed the wire back and forth to distribute it evenly.
I had originally thought of using the gear driven lathe carriage to level wind evenly but then realized that although it would work, stopping and reversing would take a lot of time when you are dealing with 16,000 turns. I would estimate that this transformer if mechanically level wound would only get 330 turns per layer and have to be done at a much lower rpm. It would be no problem with a commercial coil winding machine but beyond my budget.
I guide the wire with my fingers and try to get it as even as possible. I just try to do the best I can and live with the results. This may be critical on Hi Fi audio transformers but these old 1920’s sets seem to be forgiving as noticed by the many different transformers that have been used sucessfully as replacements.