I don't think we can know when they first requested access, but it definitely was not February 10th, since they visited Whuan in January.
According to this official "Mission Summary" (found by Denis de Bernardy), the WHO conducted a field visit to Wuhan on January 20-21 where they identified and reported evidence of human-to-human transmission of the virus in China:
The delegation visited the Wuhan Tianhe Airport, Zhongnan hospital, Hubei provincial CDC, including the BSL3 laboratory in China’s Center for Disease Control (CDC). The delegation observed and discussed active surveillance processes, temperature screening at the airport, laboratory facilities, infection prevention and control measures at the hospital and its associated fever clinics, and the deployment of the rRT-PCR test kit to detect the virus.
Data collected through detailed epidemiological investigation and through the deployment of the new test kit nationally suggests that human-to-human transmission is taking place in Wuhan. More analysis of the epidemiological data is needed to understand the full extent of human-to-human transmission. WHO stands ready to provide support to China to conduct further detailed analysis.
Mission summary: WHO Field Visit to Wuhan, China 20-21 January 2020 - Published: 22 January 2020
So claims that WHO was only allowed to visit China in February are false, though the February visit was likely a more in-depth investigation than the January 20th visit. At the same time, claims that they covered up human-to-human transmission, or were barred from reporting it by China, are also false.
This is different from their January 9th statement, where they reported based on data provided by Chinese public health officials. In that statement, they express a willingness to visit to "investigate and respond", but don't say whether they've placed an official request:
In the coming weeks, more comprehensive information is required to understand the current status and epidemiology of the outbreak, and the clinical picture. Further investigations are also required to determine the source, modes of transmission, extent of infection and countermeasures implemented. WHO continues to monitor the situation closely and, together with its partners, is ready to provide technical support to China to investigate and respond to this outbreak.
WHO Statement regarding cluster of pneumonia cases in Wuhan, China - Published: 9 January 2020