I have tried so many things. Resetting SMC and PRAM as well. Even installing Catalina again on another partition with new user to check if the problem is with some system preferences in my account (as one Senior Apple Support advisor told me to do).
Your device isn't recognized by your Mac. Follow these steps if your mouse, keyboard, or trackpad. Top comment ' Love the TouchPad.I bought this to go with a new HP Chromebox, which comes with no peripherals, and I wanted a combo device to use wirelessly from the couch, as I wanted to use it to watch over the TV, and it worked without any setup hassles, which was a worry with a wireless device connecting to a non-windows 'computer', but it worked first time out'. The Logitech K600 tops the list of the best wireless keypads this time.
What would help if someone at Apple would check my bug report about it. I have sent one through Feedback Assistant app on macOS. I have explained there that it is probably some bug with Catalina. When WiFi is ON it hangs the keyboard and trackpad. When it's OFF - keyboard and trackpad work fine. Also, I have found out that not all the WiFi networks make this problem.
So, it would be more helpful if some engineer would check and answer my bug report. I really don't understand why bug reports are so often not answered at all or being answered after many months. You know, Apple's users/customers are trying to help solve the issues. And later no-one at Apple reaches out to them.