If the patch was downloaded/installed as a result of a Patch Manager Update Management (or Update Management Wizard) task, then the download time and installation time will be recorded in the Task Details History for that task.
If the patch was downloaded/installed by the Windows Update Agent as part of a regular event cycle, that event data is reported to the WSUS server, but it is transient. It can be viewed on the "Update Events" tab of the WSUS node of the console for an indeterminate and variable period of time after the event, and can be exported from that display. However, download events are notorious for rapidly disappearing after installation events have been reported.
Additionally, the download and installation events are recorded in the client's Application Event Log (also transient though) and permanently in the ReportingEvents.log which is found in the %windir%\SoftwareDistribution folder. The WUA events recorded in the Application Event Log can be inventoried via the Managed Computer Inventory task using the Windows Update -> Event Log option (which is disabled by default). Note, though, that the Managed Computer Inventory only takes what is currently in the event log; there is no historical storage of event log data in the Patch Manager server. There is no capability for remote access to the ReportingEvents.log.