I am looking for a keyboard (and optionally a mouse) that is Bluetooth-enabled and backlit and can switch between different Bluetooth devices (pc and tablet/phone).
Any of the following Logitech Keyboards with their "Easy-Switch" technology will allow you to switch between multiple devices via Bluetooth. They have three buttons or a 3-way dial to switch instantly between up to three devices.
I personally use the tenkeyless (TKL) Logitech K480 at work since it can switch between 3 devices and has a tray for my cell phone. If you'd like a numberpad, the Logitech K780 is a good choice. Here are all of the keyboards that Logitech sells direct-to-consumer that contain the Easy-Switch technology. For some reason the K480 is on its own page here. Most Logitech keyboards can also be purchased online or at Brick-and-Mortar stores such as Amazon, Best Buy, Micro Center, Newegg, Staples, Walmart, ect.
As far as mice go, Logitech continues with their MX Anywhere lineup. Any of the MX 2S or 3 series mice will have Easy-Switch. They also let you copy files on one PC and paste them on another which is a nice feature to have. You may find the MX Anywhere 2S here.