As soon as an inverter that is supplying 120V AC power receives 120 V AC power the transfer is as fast as it takes a relay or some similar electric device to switch. You are right in saying it might be milliseconds.

There is a delay in charging however. I can't measure it, but when I turn the charging inverter off, the one that has been sitting idle picks up the charging responsibility, but there is a delay of a second or three.

If I am correct we also have delays in our transfer switch so as we go from power source to power source, or even device to device there are gaps.

You do not need to get at your batteries. Go to the electric bay and hook your voltmeter to one of the buss bars and a ground to measure battery voltage. Better yet, look at the house voltmeter on the dash.