What would be a way to turn OFF load A (10 amps) as soon as load B (10 amps as well) is turned ON, faster than the time it takes for a circuit breaker to trip (~15 ms)? Something like a voltage activated switch or relay?
If you are the one who has control over turning off load A, then a relay is a good way to do it. A DPDT relay with load A on the normally closed toggle and load B on the normally open toggle would work quite nicely. Load B would be turned on at the same instant load A is turned off, when your relay is activated.
You could achieve a changeover with a SPDT BREAK BEFORE MAKE relay if the neutral is common to both loads.
Load A is connected to the normally closed connection of the relay. Only when a switch is operated will the relay be powered and switch over to load B. By using a break before make relay the loads will never be connected at the same time. For a more sophisticated system you could replace the switch operating the relay with any appropriate electronic version. You may even like to consider a zero-crossing circuit so that the relay is operated when the load current is minimum.