If your keyboard K650 does not reconnect after a reboot at the login screen and only reconnects after the login, this might be related to BitLocker Drive Encryption which is a data protection feature that is usually enabled by IT managers enable on enterprise PCs or is enabled by default from manufacturers.

When Bitlocker is enabled, Bluetooth mice and keyboards will only re-connect after login. If BitLocker is enabled, you will be requested a key before the usual Windows Login screen or on some specific scenarios (insert a removable device (USB), detect an attack, upgrade the FW, etc.), to which your bluetooth keyboard and mouse will not be reconnected either. 

Potential solutions:

• If your Logitech device came with a Logi Bolt USB receiver, using it will solve the issue.
• Use your laptop keyboard to login
• Use an alternate keyboard connected through a USB dongle to login.

If you are allowed to disable it (please previously check with your IT Manager) you can access to your computer settings and follow these steps:

Control panel > System and Security > BitLocker Drive Encryption

Please note you will need to have a BitLocker disabling key (48-digit numerical password) that can be used to unlock your system.