Does the laptop not have a built in charger? Thus you will have nothing to do there, other than wait and something like 19.5 V to play with once charged?
The robot will not be moving while tethered, so that is easy. You know how long it takes to charge the laptop or monitor the colour of the charge lamp etc. So do that first.
disconnect the motor power, Then switch the supply to your bat charger. LiPo need special chargers, as have you have found! (and can be dangerous!)
Basically you need to check the battery voltage from time to time until it reaches the required battery voltage or until your specific charger, gives a signal that float mode has been reached (etc). This may be an LED, and then either inform you (break tether), or else switch (bat) power back to motors and trundle off.