Chinese authorities at some level -- it's not clear that this was CCP or local Wuhan decisionmakers -- clearly tried to shut down initial reports of the outbreak, as evidenced by Dr. Li Wenliang's detention and citation.

As a new disease, with the first cases possibly being acquired as early as late November 2019, it wasn't until mid to late January 2020 that city and national authorities appeared to start taking the outbreak seriously.

I'm not aware of other novel outbreaks outside of China and how quickly those have attracted notice, though there's the case of the 1976 initial outbreak of Legionaire's Disease, reported within a week of first outbreak (though the disease was both fast-acting and rapidly lethal for many victims). However epidemiologists also realised that there had been previous, unrecognised outbreaks, dating to the 1950s. See:

https://en.wikipedia.org/wiki/1976_Philadelphia_Legionnaires...

The case of HIV/AIDS in the US and elsehwere would be a case of a disease with ascribed social and moral assocations, for which early responses were very much delayed, on of several examples of stigma associated with the disease:
https://en.wikipedia.org/wiki/Discrimination_against_people_...

Reponse to the 2012 MERS outbreak appears to have been swift -- in November of 2012:
https://en.wikipedia.org/wiki/2012_Middle_East_respiratory_s...

The 2003 SARS epidemic, also emerging in China, also first emerged in November, but was not internationally reported until the following March:

https://en.wikipedia.org/wiki/Severe_acute_respiratory_syndr...

(The fact that SARS, MERS, and 2019-nCoV all appear to have emerged during November may be significant.)

Point being:

- China are slow to detect, report, and respond to new contagious epidemics. Notably so in the case of SARS, and less so, but still, with 2019-nCoV.

- Other countries, including the US, have had similar failures.

- Early unofficial public disclosure of 2019-nCoV was prosecuted by Chinese authorities, though at what direction is unclear.

- Trust between the people and the government is somewhat frayed.

- There are also numerous actors with incentives to make the most of any missteps of the government, or to inflate distrust.

Several of those groups have legitimate gripes against the Chinese government, and I've seen reports associated with Falun Gong, Hong Kong, and Taiwan specifically. I've not seen any associated with Tibet or Uygher populations. And there are various rabble-rousing actors elsewhere, doomers, and conspiracists (generally of the nutcake variety).

This ... clouds information. Many of the criticisms are legimate, some are not, and sorting fact from fiction (both between and within reports) is challenging. China are in the position where both locked-down and unfettered communications are potentially harmful -- the first for preventing accurate information from being available to both public and government, the latter for allowing agents provacateurs to rumourmonger, exaggerate bad news, and spread active disinforation. This is strong shades of attacks the West has seen over the past several years (within the US, UK, EU, and elsewhere), some of it from China itself.

How China responds will be interesting.