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I think red tape accurately and succinctly communicates the situation you describe, especially given that in a prior version you yourself wrote
Attempts to rectify this situation working through official channels did not work
which it seems to me is the very essence of (extreme) red tape. If you like we can say instead official indifference or unresponsiveness, but I don't see how that better informs the reader.
I also don't see how your wording makes the point you mention re success in decrypting and the importance of that success. Obviously a request for more resources will try to convince the recipient that such will help the writers to better do what they've been tasked to do. To spell out that their desire was ""to achieve the productivity that was possible"" would be completely superflous  of course that was the goal. And surely no reader needs to be told that better productivity / faster decrypting / etc. meant the entire difference between valuable intelligence and useless retrospective. Your wording doesn't mention any of that stuff anyway.
Further (as I've said before) I also don't see why it matters what specific type of resource was lacking  the important points are: (a) the codebreakers felt that they could not do the best job possible without more resources; (b) Churchill unhesitatingly granted them carte blanche, showing the importance he attached to their work and the breadth of his authority in making such decisions. What doesn't matter is just what it was that was lacking  typists? paperclips? light bulbs?
Some of these elements could be incorporated to enliven the narrative e.g. some kind of ""for want of a nail""-type angle  ""Look, Prime Minister, you've gathered all these first-class minds here but we can't get anything done because we have to do our own bloody TYPING!""  if sources lay something like that out.
Am I missing something?    P.S. What do you mean about WP:CIV?"