Actually, it would surprise me very much if Louis Mountbatten¨s dairy contains any information that is not available to the public already.
As far as I remember we have known since I was a teenager in the 1960s that Lady Mountbatten had an affair with Nehru, that both Mountbattens were pally with the Duchess of Argyll who was a by-word for scandalous conduct in Scotland, that Louis tried with varying degrees of success or failure to keep Princess Margaret out of trouble, and that he had a generation earlier said that George VI was the safer bet, as Edward VIII was far too pally with Mosely.
Be that all as it may, most historical archives ban the publication or access by the public to documents felt to be of a sensitive nature, either for the country, or for the family of whoever the papers had belonged to or were about.
How long the injuction is in place can vary according to circumstance.
This being so, it makes no difference whether the papers have been paid for by taxpayers' money providing the funding for their purchase, given as a gift by the original owner, or purchase by private funding.