Business Secretary Greg Clark confirmed last year that he had written to the Japanese firm to offer guarantees on how tariff-free trade could continue.
The UK government refused to provide details, claiming it had to respect commercial confidentiality.
The SNP said ministers had "flagrantly" breached freedom of information law.
Nissan announced in October last year that it planned to build two new models at its Sunderland plant.
Previously, it had expressed concern about future tariffs arising from Britain's decision to leave the EU.
The SNP submitted a Freedom of Information (FOI) request last October, asking to see the Nissan letter.
But despite being required to respond by 28 November, the Department for Business has yet to provide a response.
The SNP has now submitted an official complaint to the Information Commissioner's Office, calling on the body to urgently probe the UK government over its "desperate attempt to avoid public scrutiny".
SNP MSP Richard Lochhead said: "The Tories obviously think that if they keep on dithering and delaying then people will forget all about their sweetheart deal with Nissan - but this is fundamentally important.
"The UK government gave written assurances to an international company - one part-owned by the French government - but have refused to tell the public what these promises were.
"There is no credible reason why it should take months and months to respond to an FOI request looking for a single letter.
"The Tories are flagrantly breaching FOI law to hide their secret sweetheart deal with Nissan - and the Information Commissioner should investigate this desperate attempt to avoid public scrutiny. It's time for answers."
A UK government spokesman said the letter would only be released once its details were no longer "business-sensitive".
He added: "Our position on this has been fairly clear.
"When companies of all types and in all sectors share their investment plans with government, it is important for them to be assured that those plans will not be disclosed to their advantage.
"It's a letter to a commercial business, that by its nature contains information about that business that could be used by its competitors."
The SNP has demanded the UK government releases its correspondence with the car manufacturer Nissan to show what assurances were made ahead of Brexit.