I have this same problem with my Wireless Desktop 850 set. I did some digging which might help shed some light.
In my device manager I have a device named Keyboard and under that MS USB Dual Receiver Wireless Keyboard (similar for mouse also). I looked at the device details and saw under Power Data that the power state mappings are as such:

I did a Google search on the states (S0-S5) and mappings (D0-D5) and from my readings S0 is when the device is at full power and all keystrokes work but anything else is Sleep (device not system) and given a D2 which means:
In general, most device context is lost by the hardware. Frequently, this state preserves the part of the context that is used to signal wake events.
I'm thinking that this is the cause of the issue and when you hit the first key it signals the wake event and then once device context is regained it switches state the to S0. If someone wanted to rewrite this driver and change the state I that would eliminate
the issue as long as everything is equal to D0. I also checked on my PC and these drivers are from build 16299 (WinBuild.160101.0800).
I'm not sure why these power settings were picked in the first place. I'll assume there was good reason
or intention at Microsoft for this. If someone can explain that to me that would be cool but in my mind changing this would eliminate the problem.
One other note. I
forced Windows to update the driver (using search automatically so that it would pull from Windows Update) and it did update to Wireless Keyboard Filter Device. It did not find a Mouse driver update. I'll update this if I notice a change in performance.